#7500802

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^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, 7407998);