#6550006

order	512
nilpotency class	3
isoclinism	isoclinic class of #6533123
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 ◁ #2297 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1461 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)(17,18,23,36,22,26,21,27)(19,24,30,38,29,40,31,39)(20,32,33,25,35,42,34,41)(28,43,46,37,44,48,45,47);(2,6)(7,8)(9,10)(11,14)(12,13)(15,16)(17,19,22,29)(18,24,26,40)(20,28,35,44)(21,31,23,30)(25,37,41,47)(27,39,36,38)(32,43,42,48)(33,46,34,45);(1,3,5,4)(2,7,6,8)(10,13)(14,16)(17,20,22,35)(18,25,26,41)(19,28,29,44)(21,33,23,34)(24,37,40,47)(27,42,36,32)(30,45,31,46)(38,43,39,48);(1,4,5,3)(2,8,6,7)(9,11)(10,14)(12,15)(13,16)(18,26)(19,29)(25,41)(27,36)(28,44)(30,31)(32,42)(45,46);(11,15)(14,16)(17,21,22,23)(18,27,26,36)(19,30,29,31)(20,33,35,34)(24,38,40,39)(25,42,41,32)(28,45,44,46)(37,43,47,48)

Polycyclic presentation:

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

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, 6457202);