#7611863

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

1922 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)(3,5)(4,6)(7,8)(9,10)(11,23)(12,17)(13,18)(14,33)(15,21)(16,22)(19,32)(20,34)(24,37)(25,29)(26,39)(27,36)(28,40)(30,31)(35,38);(1,3)(2,5)(4,7)(6,8)(9,11,15,17,16,26,34,25)(10,12,20,23,22,29,21,39)(13,30,14,24,32,36,35,40)(18,37,38,31,19,28,33,27);(2,6)(3,7)(9,12,16,29)(10,17,22,25)(11,20,26,21)(13,31,32,27)(14,28,35,37)(15,23,34,39)(18,30,19,36)(24,33,40,38);(2,6)(5,8)(9,13,15,14,16,32,34,35)(10,18,21,33,22,19,20,38)(11,24,25,30,26,40,17,36)(12,27,23,37,29,31,39,28);(9,14,34,13,16,35,15,32)(10,19,21,38,22,18,20,33)(11,24,25,30,26,40,17,36)(12,28,39,31,29,37,23,27)

Polycyclic presentation:

2^9|b^2=f,c^2=i,d^2=h,e^2=fi,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,d^c=di

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, 7519059);