#6607397

order	512
nilpotency class	3
isoclinism	isoclinic class of #6574024
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	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1372 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)(19,24)(20,31)(21,26)(22,37)(23,27)(25,34)(28,44)(29,39)(30,40)(32,46)(33,43)(35,47)(36,41)(38,45)(42,48);(1,3,4,8)(2,5,6,7)(9,11)(10,13)(12,16)(14,15)(17,19)(18,24)(20,32)(21,36)(22,29)(23,30)(25,42)(26,41)(27,40)(28,35)(31,46)(33,45)(34,48)(37,39)(38,43)(44,47);(11,16)(13,14)(17,20,22,34,23,35,21,33)(18,25,26,31,27,43,37,47)(19,28,29,45,30,32,36,48)(24,38,41,44,40,42,39,46);(9,11)(10,14)(12,16)(13,15)(17,21,23,22)(18,26,27,37)(19,29,30,36)(20,33,35,34)(24,39,40,41)(25,31,43,47)(28,45,32,48)(38,46,42,44);(11,16)(13,14)(19,30)(24,40)(28,32)(29,36)(38,42)(39,41)(44,46)(45,48)

Polycyclic presentation:

2^9|a^2=f,b^2=fhi,c^2=g,d^2=i,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^d=ehi

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, 6514593);