#7519168

order	512
nilpotency class	3
isoclinism	isoclinic class of #7514040
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

2622 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7426364);