#72280968

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#346518
commutator, Frattini, cosocle	C₂×#131	C₂⁴
LCS	G ▷ C₂×#131 ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×#131 ◁ G
LECS	G ▷ C₂×#131 ▷ C₂×C₄ ▷ C₂ ▷ C₁

3962 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 64 points:

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

Polycyclic presentation:

2^10|c^2=j,d^2=eij,e^2=ij,f^2=i,g^2=j,i^2=j,c^a=ch,c^b=ch,d^a=de,d^b=dg,d^c=defij,e^a=ei,e^b=ej,e^c=ej,f^a=fi,f^b=fh,g^a=gj,g^b=gj,g^c=ghj,g^f=gj,h^d=hj,i^a=ij

Conjugacy classes & character table

67 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72281944, #72307535
maximal normal subgroup of	#72281944, #72307535

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