Finite Group Databasebeta

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#72306306

order1024
nilpotency class5
automorphism?
homologyAbelianization (H1) = C23 · Schur multiplier (H2) = C25

Aliases

No aliases for this group.

Subgroups

subgroupG/H
center, socleC2#144388
commutator, Frattini, cosocleC2×#535C23
prime radicalC2×#3021C22
LCSG ▷ C2×#535 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2 ◁ C23 ◁ C23×C4 ◁ C2×#535 ◁ G
LECSG ▷ C2×#535 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1

1566 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 1566 conjugacy classes of subgroups — too many to draw. Only its 61 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degreeμ(G) = 64
transitive degreeμt(G) = 64

A minimal faithful permutation representation — 3 generators on 64 points:

(1,2)(3,14,19,44,17,45,28,39)(4,9)(5,24,26,42)(6,10)(7,12)(8,18,32,53,33,46,31,25)(11,15,41,27)(13,29)(16,48,51,43,20,55,54,64)(21,56,58,60)(22,35)(23,37)(30,57,34,52,63,61,62,50)(36,47,40,49)(38,59);(1,3,15,46)(2,8,24,45)(4,20,47,50)(5,25,29,19)(6,28,26,53)(7,17,27,18)(9,34,56,48)(10,31,41,44)(11,39,13,32)(12,33,42,14)(16,49,52,23)(21,57,59,54)(22,51,58,61)(30,40,64,35)(36,43,38,63)(37,62,60,55);(1,4,7,23)(2,9,12,37)(3,16,17,20)(5,21,26,58)(6,22,29,59)(8,30,33,63)(10,35,13,38)(11,40,41,36)(14,43,45,64)(15,47,27,49)(18,52,46,50)(19,54,28,51)(24,60,42,56)(25,61,53,57)(31,34,32,62)(39,55,44,48)

Polycyclic presentation:

2^10|a^2=f,b^2=efgh,c^2=j,e^2=j,f^2=ij,h^2=j,b^a=be,c^a=cfij,c^b=cd,d^a=dh,d^b=dg,e^a=eghi,e^b=eghj,e^c=eg,e^d=eij,f^b=fghi,f^c=fij,f^d=fij,f^e=fi,g^a=gj,g^b=gj,h^a=hi,h^b=hij,h^c=hij,h^d=hj,h^e=hj,i^b=ij

Conjugacy classes & character table

46 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72306318, #72306348, #72306394, …
maximal normal subgroup of#72306318, #72306348, #72306394, …

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from its polycyclic presentation:

gap> F := FreeGroup(IsSyllableWordsFamily, "a", "b", "c", "d", "e", "f", "g", "h", "i", "j");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 1, F.6);;
gap> SetPower(coll, 2, F.5*F.6*F.7*F.8);;
gap> SetPower(coll, 3, F.10);;
gap> SetPower(coll, 5, F.10);;
gap> SetPower(coll, 6, F.9*F.10);;
gap> SetPower(coll, 8, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.5);;
gap> SetConjugate(coll, 3, 1, F.3*F.6*F.9*F.10);;
gap> SetConjugate(coll, 3, 2, F.3*F.4);;
gap> SetConjugate(coll, 4, 1, F.4*F.8);;
gap> SetConjugate(coll, 4, 2, F.4*F.7);;
gap> SetConjugate(coll, 5, 1, F.5*F.7*F.8*F.9);;
gap> SetConjugate(coll, 5, 2, F.5*F.7*F.8*F.10);;
gap> SetConjugate(coll, 5, 3, F.5*F.7);;
gap> SetConjugate(coll, 5, 4, F.5*F.9*F.10);;
gap> SetConjugate(coll, 6, 2, F.6*F.7*F.8*F.9);;
gap> SetConjugate(coll, 6, 3, F.6*F.9*F.10);;
gap> SetConjugate(coll, 6, 4, F.6*F.9*F.10);;
gap> SetConjugate(coll, 6, 5, F.6*F.9);;
gap> SetConjugate(coll, 7, 1, F.7*F.10);;
gap> SetConjugate(coll, 7, 2, F.7*F.10);;
gap> SetConjugate(coll, 8, 1, F.8*F.9);;
gap> SetConjugate(coll, 8, 2, F.8*F.9*F.10);;
gap> SetConjugate(coll, 8, 3, F.8*F.9*F.10);;
gap> SetConjugate(coll, 8, 4, F.8*F.10);;
gap> SetConjugate(coll, 8, 5, F.8*F.10);;
gap> SetConjugate(coll, 9, 2, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);