Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C26 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4201 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,11)(3,9,12,7)(4,8,10,5)(13,14,15,16)(17,18,22,21)(19,23,28,26)(20,24,29,30)(25,31,27,32)(33,34)(35,42)(36,41)(37,40)(38,43)(39,44)(45,48)(46,47);(3,12)(7,9)(13,15)(14,16)(17,19)(18,23)(20,27)(21,26)(22,28)(24,32)(25,29)(30,31)(33,35,39,37)(34,40,44,42)(36,45,47,38)(41,43,46,48);(1,3,6,12)(2,7,11,9)(4,15,10,13)(5,16,8,14)(17,20,22,29)(18,24,21,30)(19,25,28,27)(23,31,26,32)(33,36,39,47)(34,41,44,46)(35,45,37,38)(40,43,42,48);(1,4,6,10)(2,8,11,5)(3,13,12,15)(7,16,9,14)(17,21,22,18)(19,26,28,23)(20,24,29,30)(25,31,27,32)(33,37,39,35)(34,42,44,40)(36,45,47,38)(41,43,46,48);(1,3)(2,9)(4,15)(5,14)(6,12)(7,11)(8,16)(10,13)(17,20)(18,24)(19,27)(21,30)(22,29)(23,32)(25,28)(26,31)(33,38,39,45)(34,43,44,48)(35,36,37,47)(40,46,42,41);(1,5,6,8)(2,10,11,4)(3,14,12,16)(7,15,9,13)(17,19)(18,23)(20,27)(21,26)(22,28)(24,32)(25,29)(30,31)(33,34)(35,40)(36,46)(37,42)(38,43)(39,44)(41,47)(45,48)Polycyclic presentation:
2^9|a^2=i,b^2=g,c^2=gi,d^2=gi,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8900222);