#6609010

order	512
nilpotency class	3
isoclinism	isoclinic class of #6574298
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	#2414	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1444 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)(4,8)(5,6)(9,10)(11,16)(12,14)(13,15)(17,18,21,25)(19,27,29,23)(20,31,33,47)(22,35,32,26)(24,38,39,34)(28,42,44,46)(30,37,43,41)(36,45,40,48);(2,6)(7,8)(9,11,13,12)(10,14,15,16)(17,19)(18,23)(20,28)(21,29)(22,30)(24,36)(25,27)(26,37)(31,46)(32,43)(33,44)(34,45)(35,41)(38,48)(39,40)(42,47);(1,3,5,4)(2,7,6,8)(10,15)(11,12)(17,20,32,38,21,33,22,34)(18,24,35,47,25,39,26,31)(19,28,43,48,29,44,30,45)(23,36,41,42,27,40,37,46);(1,4,5,3)(2,8,6,7)(9,12,13,11)(10,14,15,16)(19,29)(20,33)(23,27)(24,39)(30,43)(31,47)(34,38)(37,41);(11,12)(14,16)(20,33)(24,39)(28,44)(31,47)(34,38)(36,40)(42,46)(45,48)

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=fg,d^2=fi,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=di,e^b=eh,d^c=dhi,e^c=ei,e^d=eh

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, 6516206);