#6441462

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	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2296 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1845 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=fg,c^2=i,d^2=f,e^2=i,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, 6348658);