#7506490

order	512
nilpotency class	3
isoclinism	isoclinic class of #7500701
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2297 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2032 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=fg,d^2=f,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dfg,c^b=ch,d^b=df,h^b=hi,h^c=hi,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, 7413686);