#7615677

order	512
nilpotency class	3
isoclinism	isoclinic class of #7518964
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂²×D₈	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2402 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1348 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 48 points:

(1,2)(3,7)(4,8)(5,6)(9,10,12,14)(11,15,16,13)(17,18)(19,30)(20,34)(21,39)(22,41)(23,29)(24,28)(25,33)(26,38)(27,40)(31,43)(32,44)(35,46)(36,45)(37,47)(42,48);(2,6)(7,8)(11,16)(13,15)(17,19,23,33)(18,25,29,30)(20,35,38,31)(21,37,40,42)(22,32,24,36)(26,46,34,43)(27,47,39,48)(28,44,41,45);(1,3,5,4)(2,7,6,8)(9,11,12,16)(10,13,14,15)(17,20,23,38)(18,26,29,34)(19,31,33,35)(21,32,40,36)(22,42,24,37)(25,43,30,46)(27,44,39,45)(28,48,41,47);(10,14)(13,15)(17,21,22,35,23,40,24,31)(18,27,41,43,29,39,28,46)(19,20,36,42,33,38,32,37)(25,34,44,48,30,26,45,47);(1,4,5,3)(2,7,6,8)(17,22,23,24)(18,28,29,41)(19,32,33,36)(20,37,38,42)(21,35,40,31)(25,44,30,45)(26,47,34,48)(27,46,39,43)

Polycyclic presentation:

2^9|a^2=gi,b^2=i,c^2=fgi,d^2=hi,e^2=f,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cg,d^b=dh,h^b=hi,d^c=di,e^c=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, 7522873);