#6765456

order	512
nilpotency class	3
isoclinism	isoclinic class of #6682440
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1551 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 32 points:

(1,2,5,7)(3,8,4,6)(9,10)(11,13)(12,14)(15,16)(17,18,22,26)(19,27,29,24)(20,28,31,32)(21,25,23,30);(1,2)(3,6)(4,8)(5,7)(9,11)(10,13)(12,15)(14,16)(17,19,22,29)(18,24,26,27)(20,25,31,30)(21,32,23,28);(10,14)(11,15)(17,20,23,24,22,31,21,27)(18,19,25,28,26,29,30,32);(1,3,5,4)(2,6,7,8)(17,19)(18,24)(20,30)(21,32)(22,29)(23,28)(25,31)(26,27);(1,4,5,3)(2,6,7,8)(11,15)(13,16)(17,21,22,23)(18,25,26,30)(19,28,29,32)(20,27,31,24)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6672652);