Finite Group Databasebeta

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#72318223

order1024
nilpotency class5
automorphism?
homologyAbelianization (H1) = C22×C4 · Schur multiplier (H2) = C25×C4

Aliases

No aliases for this group.

Subgroups

subgroupG/H
center, socleC2#134020
commutatorC22×#41C22×C4
Frattini, cosocleC2×#535C23
prime radical#6443332C2
LCSG ▷ C22×#41 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2 ◁ C23 ◁ C23×C4 ◁ #35327 ◁ G
LECSG ▷ C2×#535 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1

2737 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 2737 conjugacy classes of subgroups — too many to draw. Only its 99 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)(3,15)(4,20)(5,26)(6,13)(7,12)(8,14)(9,19)(10,23)(11,29)(16,43)(17,47)(18,34)(21,51)(22,54)(24,39)(25,38)(27,58)(28,41)(30,42)(31,35)(32,46)(33,49)(36,57)(37,56)(40,55)(44,48)(45,62)(50,52)(53,61)(59,64)(60,63);(1,3,5,17)(2,9,11,33,14,35,42,47)(4,21,22,16)(6,31,29,59)(7,19,30,48)(8,18,28,49)(10,36,37,43,20,50,40,62)(12,15,26,44,13,34,41,64)(23,53,27,45)(24,57,55,60)(25,52,56,46)(32,39,61,58,63,38,51,54);(1,4)(2,10)(3,16,18,45)(5,27)(6,24)(7,25)(8,23)(9,32,35,63)(11,40)(12,39)(13,38)(14,20)(15,43,34,62)(17,21,49,53)(19,46,31,60)(22,28)(26,54)(29,56)(30,55)(33,61,47,51)(36,44,50,64)(37,42)(41,58)(48,52,59,57)

Polycyclic presentation:

2^10|b^2=d,c^2=gh,d^2=ij,e^2=ij,g^2=j,h^2=j,b^a=bf,c^a=cghj,c^b=cdeij,d^a=dg,d^c=dghj,e^a=egj,e^b=egh,e^c=eghj,e^d=ej,f^b=fgj,f^d=fij,f^e=fij,g^a=gj,g^b=gi,g^f=gj,h^a=hj,h^b=hij,h^f=hj,i^a=ij

Conjugacy classes & character table

52 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72318226, #72318343, #72318389, …
maximal normal subgroup of#72318226, #72318343, #72318389, …

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, 2, F.4);;
gap> SetPower(coll, 3, F.7*F.8);;
gap> SetPower(coll, 4, F.9*F.10);;
gap> SetPower(coll, 5, F.9*F.10);;
gap> SetPower(coll, 7, F.10);;
gap> SetPower(coll, 8, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.6);;
gap> SetConjugate(coll, 3, 1, F.3*F.7*F.8*F.10);;
gap> SetConjugate(coll, 3, 2, F.3*F.4*F.5*F.9*F.10);;
gap> SetConjugate(coll, 4, 1, F.4*F.7);;
gap> SetConjugate(coll, 4, 3, F.4*F.7*F.8*F.10);;
gap> SetConjugate(coll, 5, 1, F.5*F.7*F.10);;
gap> SetConjugate(coll, 5, 2, F.5*F.7*F.8);;
gap> SetConjugate(coll, 5, 3, F.5*F.7*F.8*F.10);;
gap> SetConjugate(coll, 5, 4, F.5*F.10);;
gap> SetConjugate(coll, 6, 2, F.6*F.7*F.10);;
gap> SetConjugate(coll, 6, 4, F.6*F.9*F.10);;
gap> SetConjugate(coll, 6, 5, F.6*F.9*F.10);;
gap> SetConjugate(coll, 7, 1, F.7*F.10);;
gap> SetConjugate(coll, 7, 2, F.7*F.9);;
gap> SetConjugate(coll, 7, 6, F.7*F.10);;
gap> SetConjugate(coll, 8, 1, F.8*F.10);;
gap> SetConjugate(coll, 8, 2, F.8*F.9*F.10);;
gap> SetConjugate(coll, 8, 6, F.8*F.10);;
gap> SetConjugate(coll, 9, 1, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);