#6543776

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 ◁ #2298 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1458 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:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=i,c^2=i,d^2=fgi,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,d^b=di,f^b=fi,g^b=gi,e^c=ei,f^c=fi,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, 6450972);