#9143340

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

4174 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9050536);