Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #53390 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C25 |
| commutator, Frattini, socle, cosocle | C23 | C26 |
| prime radical | #54500 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4068 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,7,4)(3,11,14,8)(5,10,6,9)(12,15,13,16)(17,18,23,26)(19,22,25,21)(20,28,30,24)(27,31,32,29)(33,34,37,40)(35,38,42,46)(36,39,44,48)(41,45,43,47);(1,3,7,14)(2,8,4,11)(5,12,6,13)(9,15,10,16)(17,19,23,25)(18,21,26,22)(20,27,30,32)(24,31,28,29)(33,35,37,42)(34,38,40,46)(36,43,44,41)(39,47,48,45);(1,4,7,2)(3,11,14,8)(5,9,6,10)(12,15,13,16)(17,20,23,30)(18,24,26,28)(19,27,25,32)(21,31,22,29)(33,36,37,44)(34,39,40,48)(35,41,42,43)(38,45,46,47);(1,5,7,6)(2,9,4,10)(3,12,14,13)(8,15,11,16)(17,21,23,22)(18,25,26,19)(20,29,30,31)(24,27,28,32)(36,44)(39,48)(41,43)(45,47);(1,5,7,6)(2,10,4,9)(3,13,14,12)(8,15,11,16)(17,22,23,21)(18,19,26,25)(20,29,30,31)(24,27,28,32)(35,42)(38,46)(41,43)(45,47);(1,6,7,5)(2,9,4,10)(3,13,14,12)(8,15,11,16)Polycyclic presentation:
2^9|a^2=gi,b^2=gi,c^2=gi,d^2=g,e^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,d^c=dgi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8011133);