#6592425

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×#548	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1671 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)(3,6)(4,5)(7,8)(9,10,12,14)(11,13,15,16)(17,18)(19,26)(20,24)(21,30)(22,25)(23,29)(27,31)(28,32);(2,5)(6,8)(9,11)(10,13)(12,15)(14,16)(17,18)(19,27)(20,30)(21,24)(22,25)(23,28)(26,31)(29,32);(11,15)(13,16)(17,19)(18,23)(20,29)(21,31)(22,28)(24,32)(25,27)(26,30);(1,3,4,7)(2,6,5,8)(9,11)(10,13)(12,15)(14,16)(19,28)(23,27)(26,32)(29,31);(11,15)(13,16)(17,20,22,21)(18,24,25,30)(19,29,28,31)(23,32,27,26)

Polycyclic presentation:

2^9|a^2=hi,d^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,e^d=ehi

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, 6499621);