#6624289

order	512
nilpotency class	3
isoclinism	isoclinic class of #6574298
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2295 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1234 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^9|a^2=fgi,b^2=f,c^2=f,d^2=fgh,e^2=hi,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,d^b=di,e^b=ehi,d^c=dh,e^c=ei,e^d=eh

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6531485);