#6432669

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×#532	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1874 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=f,d^2=fg,e^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ei,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, 6339865);