#6503124

order	512
nilpotency class	3
isoclinism	isoclinic class of #6500186
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×#537	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2297 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1622 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)(17,18,20,25,21,22,29,32)(19,24,28,30,27,31,26,23);(1,3,4,8)(2,5,6,7)(9,10)(11,15)(12,14)(13,16)(18,22)(23,30)(24,31)(25,32);(9,11)(10,13)(12,16)(14,15)(17,19)(18,23)(20,26)(21,27)(22,30)(24,32)(25,31)(28,29);(11,16)(13,15)(17,19)(18,24)(20,28)(21,27)(22,31)(23,32)(25,30)(26,29);(10,14)(13,15)(17,20,21,29)(18,25,22,32)(19,26,27,28)(23,31,30,24)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6410320);