#9434197

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4540 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=g,c^2=hi,d^2=hi,e^2=ghi,f^2=ghi,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dghi,e^d=egi

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9341393);