#6466909

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438943
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2295 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1432 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 5 generators on 48 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6374105);