#7564810

order	512
nilpotency class	3
isoclinism	isoclinic class of #7519083
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

1845 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:

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

Polycyclic presentation:

2^9|a^2=f,c^2=fg,d^2=g,e^2=f,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^c=ei,h^d=hi

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, 7472006);