#72307288

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	#8350
commutator	C₂×M₁₆	C₂³×C₄
Frattini, cosocle	C₂×#131	C₂⁴
socle	C₂	C₂×#8350
prime radical	#10577624	C₂
LCS	G ▷ C₂×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂×#131 ▷ C₂×C₄ ▷ C₂ ▷ C₁

4011 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)(3,15,21,41)(4,22)(5,26)(6,14)(7,13)(8,9)(10,18,33,27)(11,24)(12,30)(16,45,52,63)(17,53,55,64)(19,47,20,46)(23,42)(25,44)(28,40)(29,35)(31,43)(32,34)(36,54,61,49)(37,51,62,50)(38,48,39,56)(57,58)(59,60);(2,9)(10,33)(11,34)(12,35)(13,14)(15,41)(22,44)(26,43)(36,61)(37,62)(38,39)(40,42)(45,63)(46,47)(53,64)(57,59);(1,3,4,17,7,20,24,54,8,21,25,55,6,19,32,49)(2,10,11,37,14,39,22,45,9,33,34,62,13,38,44,63)(5,27,28,52,30,56,60,51,31,18,23,16,29,48,58,50)(12,41,42,61,26,46,59,53,35,15,40,36,43,47,57,64);(3,16,21,52)(10,36,33,61)(15,45,41,63)(17,48,55,56)(18,54,27,49)(19,51,20,50)(37,47,62,46)(38,53,39,64)

Polycyclic presentation:

2^10|a^2=hij,c^2=e,d^2=hij,e^2=i,f^2=i,g^2=j,h^2=j,i^2=j,b^a=bj,c^a=cf,d^c=defgij,e^a=eh,f^a=fi,f^c=fhij,f^d=fhij,f^e=fj,g^a=gj,g^c=ghi,g^d=ghij,g^e=gj,h^a=hj,h^c=hj,i^a=ij

Conjugacy classes & character table

100 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72307300, #72307633
maximal normal subgroup of	#72307300, #72307633

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