#7613487

order	512
nilpotency class	3
isoclinism	isoclinic class of #7563327
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 ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1946 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)(3,6)(4,5)(7,8)(9,10)(11,19)(12,23)(13,26)(14,18)(15,20)(16,25)(17,27)(21,35)(22,32)(24,37)(28,36)(29,34)(30,38)(31,33)(39,40);(2,5)(6,8)(11,20)(13,27)(15,19)(16,30)(17,26)(21,33)(22,34)(25,38)(28,39)(29,32)(31,35)(36,40);(1,3,4,7)(2,6,5,8)(9,11,13,20,14,22,27,34)(10,15,17,19,18,29,26,32)(12,21,25,33,24,36,38,40)(16,28,23,31,30,35,37,39);(9,12,14,24)(10,16)(11,21,22,36)(13,25,27,38)(15,28)(17,23)(18,30)(19,31)(20,33,34,40)(26,37)(29,35)(32,39);(2,5)(6,8)(12,24)(16,30)(21,36)(23,37)(25,38)(28,35)(31,39)(33,40)

Polycyclic presentation:

2^9|c^2=fg,d^2=hi,g^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,g^a=gi,h^a=hi,c^b=cg,d^b=dh,g^b=gi,e^d=ei

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, 7520683);