#6580558

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573451
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 ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1569 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)(3,8)(4,6)(5,7)(9,10)(11,13)(12,14)(15,16)(17,18,22,27)(19,24,32,42)(20,33,35,25)(21,26,31,41)(23,28,30,40)(29,47,48,39)(34,45,37,43)(36,46,38,44);(1,3,5,4)(2,6,7,8)(9,11,12,15)(10,13,14,16)(17,19)(18,24)(20,34)(21,30)(22,32)(23,31)(25,43)(26,40)(27,42)(28,41)(29,36)(33,45)(35,37)(38,48)(39,44)(46,47);(10,14)(13,16)(17,20,22,35)(18,25,27,33)(19,29,32,48)(21,37,31,34)(23,38,30,36)(24,39,42,47)(26,45,41,43)(28,46,40,44);(1,4,5,3)(2,6,7,8)(17,21,22,31)(18,26,27,41)(19,30,32,23)(20,29,35,48)(24,40,42,28)(25,39,33,47)(34,36,37,38)(43,44,45,46);(19,31)(21,32)(23,30)(24,41)(26,42)(28,40)(29,34)(36,38)(37,48)(39,43)(44,46)(45,47)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6487754);