#7508523

order	512
nilpotency class	3
isoclinism	isoclinic class of #7500695
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
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1555 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)(3,4)(5,7)(6,8)(9,10)(11,15)(12,14)(13,16)(19,27)(24,31)(28,29)(30,32);(1,3)(2,6)(4,7)(5,8)(17,18)(19,24)(20,25)(21,23)(22,26)(27,31)(28,30)(29,32);(9,11,12,16)(10,13,14,15)(18,23)(20,21)(24,30)(25,26)(28,29)(31,32);(2,7)(3,8)(10,14)(13,15)(17,19)(18,24)(20,29)(21,28)(22,27)(23,30)(25,32)(26,31);(1,4,5,6)(2,3,7,8)(17,20,22,21)(18,25,26,23)(19,28,27,29)(24,30,31,32)

Polycyclic presentation:

2^9|c^2=g,e^2=fi,h^2=i,b^a=bf,c^a=cg,d^a=dfgi,c^b=ch,d^b=df,h^b=hi,e^c=ei,h^c=hi,e^d=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7415719);