#6506584

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 ◁ #2297 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1632 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)(22,37)(23,28)(25,31)(27,38)(30,45)(32,39)(33,40)(34,41)(35,43)(36,48)(42,47)(44,46);(2,5)(6,8)(9,10)(11,15)(12,14)(13,16)(17,19,22,20,23,31,38,33)(18,24,27,29,28,40,37,25)(21,34,36,47,35,30,46,32)(26,42,44,41,43,39,48,45);(1,3,4,7)(2,6,5,8)(9,11)(10,13)(12,16)(14,15)(17,20,22,31,23,33,38,19)(18,25,27,24,28,29,37,40)(21,32,36,34,35,47,46,30)(26,41,44,39,43,45,48,42);(11,16)(13,15)(17,21,23,35)(18,26,28,43)(19,30,31,34)(20,32,33,47)(22,36,38,46)(24,39,40,42)(25,41,29,45)(27,44,37,48);(10,14)(13,15)(21,35)(26,43)(30,34)(32,47)(36,46)(39,42)(41,45)(44,48)

Polycyclic presentation:

2^9|a^2=fgi,b^2=g,c^2=f,d^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,d^b=di,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, 6413780);