#7614044

order	512
nilpotency class	3
isoclinism	isoclinic class of #7518944
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 ◁ #2397 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2627 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)(11,21)(12,17)(13,16)(14,28)(15,20)(18,23)(19,29)(22,35)(24,32)(25,30)(26,39)(27,31)(33,36)(34,38)(37,40);(2,4)(6,7)(9,11,14,13,15,23,29,27)(10,16,19,21,20,31,28,18)(12,22,26,25,24,36,38,37)(17,30,34,35,32,40,39,33);(5,6,8,7)(9,12,15,24)(10,17,20,32)(11,22,23,36)(13,25,27,37)(14,26,29,38)(16,30,31,40)(18,33,21,35)(19,34,28,39);(1,2,3,4)(5,7,8,6)(12,24)(17,32)(22,36)(25,37)(26,38)(30,40)(33,35)(34,39);(9,13,14,23,15,27,29,11)(10,18,19,16,20,21,28,31)(12,25,26,36,24,37,38,22)(17,33,34,30,32,35,39,40)

Polycyclic presentation:

2^9|b^2=f,c^2=gi,d^2=h,e^2=f,f^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,f^a=fi,c^b=cg,d^b=dh,e^b=ei,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, 7521240);