#6580747

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×#548	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1557 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)(17,18,21,23)(19,24,29,37)(20,30,34,25)(22,26,35,38)(27,40,36,39)(28,43,32,41)(31,42,33,44)(45,47,46,48);(1,3,5,4)(2,6,7,8)(9,11,12,15)(10,13,14,16)(18,23)(19,27)(20,31)(22,35)(24,39)(25,42)(29,36)(30,44)(33,34)(37,40)(41,43)(45,46);(10,14)(13,16)(17,19)(18,24)(20,32)(21,29)(22,36)(23,37)(25,43)(26,39)(27,35)(28,34)(30,41)(31,46)(33,45)(38,40)(42,48)(44,47);(1,4,5,3)(2,6,7,8)(19,27)(20,33)(22,35)(24,40)(25,42)(26,38)(28,32)(29,36)(30,44)(31,34)(37,39)(41,43);(17,20,21,34)(18,25,23,30)(19,28,29,32)(22,33,35,31)(24,41,37,43)(26,42,38,44)(27,45,36,46)(39,47,40,48)

Polycyclic presentation:

2^9|a^2=i,b^2=fgi,d^2=fi,e^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,e^b=eh,h^b=hi,d^c=dhi,e^c=ei,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, 6487943);