#6519533

order	512
nilpotency class	3
isoclinism	isoclinic class of #6500456
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2298 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1491 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:

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

Polycyclic presentation:

2^9|a^2=fgi,b^2=g,c^2=f,d^2=h,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,e^b=ehi,d^c=dhi,e^d=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6426729);