#7508529

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
prime radical	#2535	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1469 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)(18,23)(24,32)(25,28)(26,31);(1,3)(2,6)(4,7)(5,8)(17,18)(19,25)(20,28)(21,23)(22,31)(24,27)(26,30)(29,32);(9,11,12,16)(10,13,14,15)(17,19,21,20)(18,24,23,32)(22,29,30,27)(25,26,28,31);(2,7)(3,8)(10,14)(13,15)(17,20,21,19)(18,25,23,28)(22,29,30,27)(24,26,32,31);(1,4,5,6)(2,3,7,8)(19,20)(24,32)(25,28)(27,29)

Polycyclic presentation:

2^9|c^2=gi,d^2=i,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, 7415725);