#7500986

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

3277 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)(9,10,13,18,14,19,27,17)(11,20,24,29,23,33,22,15)(12,16,25,31,26,32,38,30)(21,34,37,39,36,40,35,28);(5,6)(7,8)(9,11)(10,15)(12,21)(13,22)(14,23)(16,28)(17,20)(18,33)(19,29)(24,27)(25,35)(26,36)(30,34)(31,40)(32,39)(37,38);(2,4)(6,8)(9,12,14,26)(10,16,19,32)(11,21,23,36)(13,25,27,38)(15,28,29,39)(17,30,18,31)(20,34,33,40)(22,35,24,37);(2,4)(10,17)(11,22)(13,27)(16,30)(18,19)(20,33)(21,35)(23,24)(25,38)(31,32)(34,40)(36,37);(11,23)(15,29)(20,33)(21,36)(22,24)(28,39)(34,40)(35,37)

Polycyclic presentation:

2^9|a^2=f,c^2=i,f^2=i,b^a=bf,c^a=cg,d^a=dfg,c^b=ch,d^b=df,e^b=ei,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, 7408182);