#6548791

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×D₄	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1391 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:

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

Polycyclic presentation:

2^9|a^2=fgi,b^2=f,c^2=i,d^2=fi,e^2=f,g^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,d^b=di,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, 6455987);