Finite Group Databasebeta

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#72341203

order1024
nilpotency class6
automorphism?
homologyAbelianization (H1) = C22×C4 · Schur multiplier (H2) = C23

Aliases

No aliases for this group.

Subgroups

subgroupG/H
center, socleC2#150724
commutator#379C22×C4
Frattini, cosocle#1781C23
prime radicalC2×#534D4
LCSG ▷ #379 ▷ C25 ▷ C23 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2 ◁ C22 ◁ C24 ◁ C23×D4 ◁ #21749 ◁ G
LECSG ▷ #1781 ▷ C25 ▷ C23 ▷ C22 ▷ C2 ▷ C1

624 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 624 conjugacy classes of subgroups — too many to draw. Only its 53 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:

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

Polycyclic presentation:

2^10|a^2=fij,b^2=d,c^2=gh,e^2=ghj,b^a=be,c^a=cfh,c^b=cefghj,d^a=df,d^c=dfhij,e^a=eg,e^b=efghi,e^c=eghij,e^d=ehi,f^b=fh,f^c=fi,f^e=fij,g^a=gij,g^b=gij,g^c=gij,g^d=gj,g^e=gj,h^a=hi,h^c=hij,h^e=hj,i^b=ij

Conjugacy classes & character table

40 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72341206, #72341219
maximal normal subgroup of#72341206, #72341219

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