#7502135

order	512
nilpotency class	3
isoclinism	isoclinic class of #7500701
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#23386	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2280 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2672 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 40 points:

(1,2)(3,4)(9,10,14,18)(11,19,22,35)(12,23,25,38)(13,27,29,17)(15,28,31,21)(16,32,34,26)(20,36,37,30)(24,33,39,40);(5,6)(7,8)(9,11,14,22)(10,15,18,31)(12,24,25,39)(13,28,29,21)(16,33,34,40)(17,35,27,19)(20,32,37,26)(23,30,38,36);(2,4)(6,8);(2,4)(9,12,13,26,14,25,29,32)(10,16,17,23,18,34,27,38)(11,20,21,24,22,37,28,39)(15,30,19,33,31,36,35,40);(12,25)(16,34)(20,37)(23,38)(24,39)(26,32)(30,36)(33,40)

Polycyclic presentation:

2^9|a^2=i,b^2=i,d^2=f,f^2=i,b^a=bf,c^a=cg,d^a=dfg,f^a=fi,c^b=ch,d^b=df,f^b=fi,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, 7409331);