#7500806

order	512
nilpotency class	3
isoclinism	isoclinic class of #7500669
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸×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 ◁ #2280 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3270 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)(9,10,14,20,15,19,12,17)(11,21,22,32,24,35,29,16)(13,26,25,34,28,36,30,18)(23,31,27,33,39,40,38,37);(5,6)(7,8)(9,11,15,24)(10,16,19,32)(12,22,14,29)(13,27,28,38)(17,21,20,35)(18,33,34,37)(23,30,39,25)(26,31,36,40);(2,4)(6,8)(9,12,15,14)(10,17,19,20)(11,22,24,29)(13,25,28,30)(16,21,32,35)(18,26,34,36)(23,38,39,27)(31,37,40,33);(2,4)(9,13,15,28)(10,18,19,34)(11,23,24,39)(12,25,14,30)(16,31,32,40)(17,26,20,36)(21,37,35,33)(22,38,29,27);(10,19)(16,32)(17,20)(18,34)(21,35)(26,36)(31,40)(33,37)

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=i,d^2=i,f^2=i,b^a=bf,c^a=cg,d^a=dfg,e^a=ei,c^b=ch,d^b=df,f^b=fi,f^d=fi

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, 7408002);