#72307136

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#8376
commutator	C₂²×C₈	C₂³×C₄
Frattini, cosocle	C₂×#131	C₂⁴
socle	C₂	C₂×#8376
prime radical	#10576601	C₂
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂×#131 ▷ C₂×C₄ ▷ C₂ ▷ C₁

3759 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:

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

Polycyclic presentation:

2^10|a^2=hij,d^2=e,e^2=i,f^2=ij,g^2=j,h^2=j,i^2=j,b^a=bj,c^a=cj,c^b=cj,d^a=df,d^b=dfgj,e^a=ei,e^b=eh,f^a=fij,f^b=fi,f^d=fj,g^a=gj,g^b=gj,g^d=ghi,g^e=gj,h^a=hj,h^b=hj,h^d=hj,i^a=ij,i^b=ij

Conjugacy classes & character table

88 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72307146, #72307386
maximal normal subgroup of	#72307146, #72307386

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