#6466323

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

1322 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)(9,10,12,14)(11,15,16,13)(17,18)(19,24)(20,25)(21,26)(22,28)(23,27)(29,38)(30,39)(31,41)(32,40)(33,43)(34,42)(35,44)(36,45)(37,46)(47,48);(2,5)(6,8)(17,19,23,32)(18,24,27,40)(20,30,34,47)(21,35,33,29)(22,37,36,31)(25,39,42,48)(26,44,43,38)(28,46,45,41);(9,11,12,16)(10,13,14,15)(17,20,22,33,23,34,36,21)(18,25,28,43,27,42,45,26)(19,29,31,30,32,35,37,47)(24,38,41,39,40,44,46,48);(17,21,36,34,23,33,22,20)(18,26,45,42,27,43,28,25)(19,30,37,29,32,47,31,35)(24,39,46,38,40,48,41,44);(1,3,4,7)(2,6,5,8)(18,27)(24,40)(25,42)(26,43)(28,45)(38,44)(39,48)(41,46)

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=gh,d^2=hi,e^2=f,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, 6373519);