#6592095

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573615
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	C₂×#547	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1724 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:

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

Polycyclic presentation:

2^9|a^2=fh,b^2=fi,c^2=i,d^2=i,e^2=i,g^2=i,b^a=bf,c^a=cg,e^a=ei,g^a=gi,c^b=ch,d^b=di,e^b=eh,d^c=dh,g^c=gi,e^d=eh

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6499291);