#7563288

order	512
nilpotency class	3
isoclinism	isoclinic class of #7519008
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2730 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=fi,c^2=gi,d^2=g,e^2=fi,f^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,f^a=fi,c^b=cg,d^b=dh,d^c=di,e^c=ei,e^d=ei

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7470484);