#6592093

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×#546	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1811 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:

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

Polycyclic presentation:

2^9|a^2=fh,b^2=fi,c^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, 6499289);