#6493757

order	512
nilpotency class	3
isoclinism	isoclinic class of #6469971
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×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1509 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)(3,5)(4,6)(7,8)(11,15)(14,16)(17,18,23,36,22,26,21,27)(19,28,35,24,30,43,31,39)(20,25,37,48,33,41,34,42)(29,44,47,38,32,46,45,40);(1,3,4,7)(2,5,6,8)(9,10)(11,14)(12,13)(15,16)(17,19)(18,24)(20,32)(21,35)(22,30)(23,31)(25,40)(26,39)(27,43)(28,36)(29,33)(34,45)(37,47)(38,41)(42,44)(46,48);(2,6)(5,8)(10,13)(14,16)(17,20,22,33)(18,25,26,41)(19,29,30,32)(21,34,23,37)(24,38,39,40)(27,42,36,48)(28,44,43,46)(31,45,35,47);(10,13)(14,16)(19,30)(20,33)(24,39)(25,41)(28,43)(31,35)(34,37)(42,48);(9,11,12,15)(10,14,13,16)(18,26)(20,33)(24,39)(27,36)(28,43)(29,32)(34,37)(45,47)

Polycyclic presentation:

2^9|a^2=fi,b^2=g,c^2=i,e^2=hi,f^2=i,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=dh,f^b=fi,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, 6400953);