#6500801

order	512
nilpotency class	3
isoclinism	isoclinic class of #6499761
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	C₂³×D₄	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄×D₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1878 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=f,e^2=i,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,e^b=eh,d^c=dh,g^c=gi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6407997);