#7611854

order	512
nilpotency class	3
isoclinism	isoclinic class of #7563319
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

1931 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)(3,5)(4,6)(7,8)(9,10)(11,21)(12,27)(13,29)(14,32)(15,20)(16,18)(17,25)(19,28)(22,36)(23,35)(24,37)(26,33)(30,40)(31,39)(34,38);(1,3)(2,5)(4,7)(6,8)(9,11,12,25,15,26,16,22)(10,17,18,21,20,36,27,33)(13,30,14,24,28,38,31,23)(19,35,32,34,29,37,39,40);(2,6)(3,7)(9,12,15,16)(10,18,20,27)(11,22,26,25)(13,14,28,31)(17,33,36,21)(19,32,29,39)(23,38,24,30)(34,35,40,37);(2,6)(5,8)(9,13,16,31,15,28,12,14)(10,19,18,32,20,29,27,39)(11,23,25,30,26,24,22,38)(17,34,33,35,36,40,21,37);(9,14,12,28,15,31,16,13)(10,19,18,32,20,29,27,39)(11,24,25,38,26,23,22,30)(17,35,21,34,36,37,33,40)

Polycyclic presentation:

2^9|b^2=f,c^2=i,d^2=h,e^2=f,f^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,f^a=fi,c^b=cg,d^b=dh,h^b=hi

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, 7519050);