#7624987

order	512
nilpotency class	4
isoclinism	isoclinic class of #7624812
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 ◁ C₈×#41 ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1410 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:

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

Polycyclic presentation:

2^9|c^2=gh,e^2=fi,g^2=hi,h^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gh,h^a=hi,c^b=ci,d^b=df

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7532183);