#6547964

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

1535 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)(17,18,21,26,22,27,37,25)(19,28,31,42,32,23,30,39)(20,33,36,41,35,40,34,24)(29,44,45,38,47,48,46,43);(1,3,4,8)(2,5,6,7)(9,10)(11,14)(12,13)(15,16)(17,19)(18,23)(20,29)(21,31)(22,32)(24,38)(25,42)(26,39)(27,28)(30,37)(33,48)(34,46)(35,47)(36,45)(40,44)(41,43);(10,13)(14,16)(17,20)(18,24)(19,29)(21,34)(22,35)(23,38)(25,33)(26,40)(27,41)(28,43)(30,45)(31,46)(32,47)(36,37)(39,44)(42,48);(9,11)(10,14)(12,15)(13,16)(17,21,22,37)(18,25,27,26)(19,30,32,31)(20,34,35,36)(23,39,28,42)(24,33,41,40)(29,45,47,46)(38,44,43,48);(11,15)(14,16)(17,21,22,37)(18,26,27,25)(19,31,32,30)(20,34,35,36)(23,39,28,42)(24,40,41,33)(29,46,47,45)(38,44,43,48)

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,d^a=di,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, 6455160);