#6455096

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437547
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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2298 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1466 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)(11,14)(12,13)(15,16)(17,18)(19,27)(20,25)(21,23)(22,26)(24,30)(28,38)(29,43)(31,42)(32,36)(33,37)(34,40)(35,41)(39,45)(44,47)(46,48);(1,3,4,8)(2,5,6,7)(18,23)(19,28)(20,31)(22,33)(24,38)(25,41)(27,40)(30,34)(32,35)(36,42)(39,43)(44,46);(10,13)(14,16)(17,19,21,30)(18,24,23,27)(20,29,32,45)(22,34,33,28)(25,39,36,43)(26,38,37,40)(31,44,35,46)(41,47,42,48);(17,20,21,32)(18,25,23,36)(19,29,30,45)(22,35,33,31)(24,39,27,43)(26,41,37,42)(28,44,34,46)(38,47,40,48);(9,11,12,15)(10,14,13,16)(19,30)(24,27)(28,34)(29,45)(38,40)(39,43)(44,46)(47,48)

Polycyclic presentation:

2^9|a^2=fi,b^2=fi,c^2=i,d^2=i,e^2=gi,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=dh,h^b=hi,e^c=ei,h^c=hi,h^d=hi

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, 6362292);