#7502565

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

1736 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)(5,7)(6,8)(9,10)(11,17)(12,24)(13,30)(14,32)(15,20)(16,21)(18,28)(19,31)(22,38)(23,34)(25,33)(26,35)(27,37)(29,36)(39,40);(1,3)(2,6)(4,7)(5,8)(9,11,16,13)(10,17,21,30)(12,25,27,39)(14,23,15,31)(18,35,36,38)(19,32,34,20)(22,28,26,29)(24,33,37,40);(9,12,16,27)(10,18,21,36)(11,22,13,26)(14,29,15,28)(17,33,30,40)(19,35,34,38)(20,37,32,24)(23,39,31,25);(2,7)(3,8)(9,13,16,11)(10,19,21,34)(12,26,27,22)(14,23,15,31)(17,32,30,20)(18,35,36,38)(24,40,37,33)(25,29,39,28);(1,4,5,6)(2,3,7,8)(12,27)(18,36)(22,26)(24,37)(25,39)(28,29)(33,40)(35,38)

Polycyclic presentation:

2^9|b^2=i,c^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, 7409761);