#6851475

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	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂²×D₈	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2295 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1314 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:

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

Polycyclic presentation:

2^9|b^2=h,c^2=h,d^2=fi,e^2=h,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efhi,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, 6758671);