#6761289

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	#30567	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1334 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)(3,6)(4,8)(5,7)(17,18,22,23,21,26,19,24)(20,28,29,32,30,25,27,31);(1,3,6,2)(4,8,5,7)(9,10)(11,15)(12,14)(13,16)(17,19,21,22)(18,23,26,24)(20,29,30,27)(25,32,28,31);(9,11,12,16)(10,13,14,15)(18,24)(19,22)(23,26)(25,32)(27,29)(28,31);(1,4,6,5)(2,7,3,8)(17,20,21,30)(18,25,26,28)(19,27,22,29)(23,31,24,32);(1,5,6,4)(2,7,3,8)(10,14)(13,15)(20,30)(25,28)(27,29)(31,32)

Polycyclic presentation:

2^9|a^2=g,b^2=f,c^2=hi,d^2=f,e^2=fi,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, 6668485);