#6868030

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667900
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	#31021	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1533 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6775226);