#7502562

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

1765 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)(11,16)(12,25)(13,19)(14,17)(15,20)(18,26)(21,30)(22,36)(23,31)(24,32)(27,33)(28,34)(29,35)(37,40)(38,39);(1,3)(2,6)(4,7)(5,8)(9,11,15,24)(10,16,20,32)(12,22,27,38)(13,21,14,23)(17,31,19,30)(18,29,34,40)(25,36,33,39)(26,35,28,37);(10,17)(11,21)(13,14)(18,25)(19,20)(22,35)(23,24)(26,28)(30,31)(33,34)(36,39)(37,38);(2,7)(3,8)(9,12,15,27)(10,18,20,34)(11,22,24,38)(13,26,14,28)(16,29,32,40)(17,25,19,33)(21,35,23,37)(30,39,31,36);(1,4,5,6)(2,3,7,8)(9,13,15,14)(10,19,20,17)(11,21,24,23)(12,26,27,28)(16,30,32,31)(18,33,34,25)(22,35,38,37)(29,39,40,36)

Polycyclic presentation:

2^9|b^2=i,d^2=i,e^2=fi,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, 7409758);