#9159943

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

4176 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=gi,c^2=ghi,d^2=hi,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, 9067139);