#6432683

order	512
nilpotency class	3
isoclinism	isoclinic class of #6431710
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#535	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1942 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=fi,b^2=f,c^2=i,d^2=fg,e^2=i,h^2=i,b^a=bf,c^a=cg,c^b=ch,h^b=hi,h^c=hi,e^d=ei

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6339879);