#7611865

order	512
nilpotency class	3
isoclinism	isoclinic class of #7563333
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

1905 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)(11,14)(12,19)(13,20)(15,31)(16,21)(17,22)(18,24)(23,33)(25,32)(26,35)(27,39)(28,40)(29,37)(30,38)(34,36);(1,3)(2,5)(4,7)(6,8)(9,11,15,24,17,25,16,20)(10,18,21,14,22,13,31,32)(12,23,28,36,30,39,29,35)(19,34,37,33,38,26,40,27);(2,6)(3,7)(9,12)(10,19)(11,23)(13,26)(14,33)(15,28)(16,29)(17,30)(18,34)(20,35)(21,37)(22,38)(24,36)(25,39)(27,32)(31,40);(2,6)(5,8)(9,13,16,14,17,18,15,32)(10,20,21,11,22,24,31,25)(12,26,29,33,30,34,28,27)(19,35,37,23,38,36,40,39);(9,14,15,13,17,32,16,18)(10,20,21,11,22,24,31,25)(12,27,28,34,30,33,29,26)(19,36,37,39,38,35,40,23)

Polycyclic presentation:

2^9|b^2=f,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,e^c=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, 7519061);