#6455363

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437537
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

1484 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)(3,6)(4,5)(7,8)(9,10)(11,14)(12,13)(15,16)(17,18)(19,24)(20,26)(21,36)(22,25)(23,27)(28,39)(29,38)(30,37)(31,40)(32,42)(33,44)(34,41)(35,43)(45,48)(46,47);(2,5)(6,8)(17,19,22,31)(18,24,25,40)(20,32,34,46)(21,30,23,28)(26,42,41,47)(27,39,36,37)(29,33,45,35)(38,44,48,43);(1,3,4,7)(2,6,5,8)(10,13)(14,16)(18,25)(19,28)(21,23)(24,37)(26,41)(29,32)(30,31)(33,35)(38,47)(39,40)(42,48)(45,46);(17,20,22,34)(18,26,25,41)(19,29,31,45)(21,35,23,33)(24,38,40,48)(27,44,36,43)(28,32,30,46)(37,47,39,42);(9,11,12,15)(10,14,13,16)(17,21,22,23)(18,27,25,36)(19,30,31,28)(20,33,34,35)(24,39,40,37)(26,43,41,44)(29,32,45,46)(38,47,48,42)

Polycyclic presentation:

2^9|b^2=i,c^2=f,d^2=i,e^2=g,h^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,d^b=dh,h^b=hi,e^c=ei,h^c=hi,e^d=ei,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, 6362559);