#6597496

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573615
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

1655 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)(19,23)(20,30)(21,33)(22,25)(24,31)(26,35)(27,41)(28,37)(29,45)(32,40)(34,39)(36,43)(38,44)(42,48)(46,47);(1,3,4,8)(2,5,6,7)(9,11)(10,13)(12,16)(14,15)(17,19)(18,23)(20,27)(21,29)(22,28)(24,36)(25,37)(26,38)(30,41)(31,43)(32,42)(33,45)(34,46)(35,44)(39,47)(40,48);(11,16)(13,14)(17,20,22,32)(18,24,25,39)(19,27,28,42)(21,34,35,31)(23,36,37,47)(26,40,33,30)(29,46,44,43)(38,48,45,41);(9,11)(10,14)(12,16)(13,15)(18,25)(19,28)(24,39)(26,33)(27,42)(29,44)(30,40)(43,46);(11,16)(13,14)(19,28)(23,37)(27,42)(29,44)(36,47)(38,45)(41,48)(43,46)

Polycyclic presentation:

2^9|a^2=f,b^2=fh,c^2=i,g^2=i,b^a=bf,c^a=cg,d^a=di,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

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6504692);