#6548508

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	C₂×#548	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1540 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:

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

Polycyclic presentation:

2^9|a^2=fgi,b^2=fi,d^2=fi,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=di,c^b=ch,d^b=di,e^b=ei,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, 6455704);