#7502561

order	512
nilpotency class	3
isoclinism	isoclinic class of #7500736
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×D₄²
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1785 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|b^2=i,d^2=i,e^2=f,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dfg,g^a=gi,c^b=ch,d^b=df,h^b=hi,e^c=ei,g^c=gi,h^c=hi,g^d=gi

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, 7409757);