#7563232

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

Polycyclic presentation:

2^9|a^2=i,b^2=fi,c^2=g,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^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, 7470428);