#7623139

order	512
nilpotency class	4
isoclinism	isoclinic class of #7622825
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂³×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁵
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×#41 ◁ C₄×#131 ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1532 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)(3,6)(4,5)(7,8)(9,10)(11,22)(12,17)(13,19)(14,20)(15,29)(16,32)(18,24)(21,31)(23,35)(25,33)(26,37)(27,38)(28,39)(30,36)(34,40);(2,5)(6,8)(9,11,14,12)(10,17,20,22)(13,25,16,23)(15,24,28,26)(18,29,37,39)(19,35,32,33)(21,34,38,36)(27,40,31,30);(9,11,14,12)(10,18,20,37)(13,23,16,25)(15,30,28,40)(17,29,22,39)(19,36,32,34)(21,33,38,35)(24,27,26,31);(1,3,4,7)(2,6,5,8)(9,12,14,11)(10,17,20,22)(13,23,16,25)(15,26,28,24)(18,29,37,39)(19,35,32,33)(21,34,38,36)(27,30,31,40);(9,13,14,16)(10,19,20,32)(11,23,12,25)(15,27,28,31)(17,33,22,35)(18,36,37,34)(21,29,38,39)(24,30,26,40)

Polycyclic presentation:

2^9|b^2=i,c^2=i,d^2=f,e^2=i,g^2=hi,h^2=i,b^a=bf,c^a=cg,g^a=gh,h^a=hi,e^b=ei,g^c=gh,h^c=hi,e^d=ei

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7530335);