#6597327

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573591
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

1687 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=fhi,c^2=i,d^2=i,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ei,g^a=gi,c^b=ch,d^b=di,e^b=ehi,d^c=dh,g^c=gi,e^d=eh

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6504523);