#6466384

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438991
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

1318 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)(17,18,23,28)(19,29,31,24)(20,33,32,25)(21,26,38,44)(22,27,34,40)(30,45,47,39)(35,42,36,41)(37,43,46,48);(2,5)(6,8)(17,19,23,31)(18,24,28,29)(20,34,32,22)(21,36,38,35)(25,40,33,27)(26,42,44,41)(30,46,47,37)(39,48,45,43);(9,11,12,16)(10,13,14,15)(17,20,22,19,23,32,34,31)(18,25,27,24,28,33,40,29)(21,35,37,30,38,36,46,47)(26,41,43,39,44,42,48,45);(17,21,34,46,23,38,22,37)(18,26,40,48,28,44,27,43)(19,30,20,35,31,47,32,36)(24,39,25,41,29,45,33,42);(1,3,4,7)(2,6,5,8)(17,22,23,34)(18,27,28,40)(19,20,31,32)(21,37,38,46)(24,25,29,33)(26,43,44,48)(30,35,47,36)(39,41,45,42)

Polycyclic presentation:

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

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, 6373580);