#6547932

order	512
nilpotency class	3
isoclinism	isoclinic class of #6533195
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	C₂×#547	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1531 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6455128);