#6776563

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667932
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2295 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1169 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 56
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 5 generators on 56 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=fi,c^2=i,d^2=hi,e^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efh,e^c=ei,g^c=gi,e^d=ei

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6683759);