#6867388

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	#31042	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1528 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)(3,7)(4,6)(5,8)(17,18,23,32,22,26,21,25)(19,27,28,24,30,31,29,20);(1,3)(2,5)(4,8)(6,7)(9,10)(11,15)(12,16)(13,14);(10,14)(15,16)(17,19)(18,24)(20,26)(21,28)(22,30)(23,29)(25,31)(27,32);(2,6)(5,7)(9,11,13,12)(10,15,14,16)(17,20,22,24)(18,19,26,30)(21,31,23,27)(25,29,32,28);(3,8)(5,7)(9,12,13,11)(10,15,14,16)(17,21,22,23)(18,25,26,32)(19,28,30,29)(20,27,24,31)

Polycyclic presentation:

2^9|a^2=gi,d^2=hi,e^2=hi,g^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,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, 6774584);