#6543712

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

1460 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)(19,28,30,40)(20,32,33,42)(21,25,37,26)(23,36,39,31)(24,34,41,35)(29,38,46,48)(43,44,47,45);(2,5)(6,8)(9,10)(11,14)(12,13)(15,16)(17,19,22,30)(18,23,27,39)(20,29,33,46)(21,36,37,31)(24,38,41,48)(25,28,26,40)(32,43,42,47)(34,45,35,44);(10,13)(14,16)(17,20,22,33)(18,24,27,41)(19,29,30,46)(21,34,37,35)(23,38,39,48)(25,32,26,42)(28,43,40,47)(31,44,36,45);(1,3,4,7)(2,6,5,8)(9,11)(10,14)(12,15)(13,16)(19,30)(20,33)(23,39)(24,41)(28,40)(31,36)(32,42)(34,35);(11,15)(14,16)(17,21,22,37)(18,25,27,26)(19,31,30,36)(20,34,33,35)(23,40,39,28)(24,32,41,42)(29,44,46,45)(38,47,48,43)

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,e^b=ei,f^b=fi,g^b=gi,d^c=di,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, 6450908);