#6581403

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573511
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 ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1539 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)(3,7)(4,6)(5,8)(9,10)(11,13)(12,14)(15,16)(17,18,22,27)(19,29,31,24)(20,25,23,28)(21,26,35,41)(30,42,36,37)(32,43,45,38)(33,39,34,40)(44,48,46,47);(1,3,5,4)(2,6,8,7)(9,11,12,15)(10,13,14,16)(19,30)(20,23)(21,34)(24,37)(25,28)(26,40)(29,42)(31,36)(32,45)(33,35)(38,43)(39,41);(1,3,5,4)(2,7,8,6)(10,14)(13,16)(17,19)(18,24)(20,30)(21,32)(22,31)(23,36)(25,37)(26,38)(27,29)(28,42)(33,44)(34,46)(35,45)(39,47)(40,48)(41,43);(1,4,5,3)(2,6,8,7)(17,20)(18,25)(19,31)(21,35)(22,23)(24,29)(26,41)(27,28)(32,44)(38,47)(43,48)(45,46);(17,21,22,35)(18,26,27,41)(19,32,31,45)(20,33,23,34)(24,38,29,43)(25,39,28,40)(30,44,36,46)(37,47,42,48)

Polycyclic presentation:

2^9|a^2=i,b^2=fgi,c^2=f,d^2=f,e^2=i,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=di,e^b=ehi,h^b=hi,d^c=dhi,h^c=hi,e^d=eh,h^d=hi,h^e=hi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6488599);