Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #35346 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
| direct product | C2×#35629 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C23×#41 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
5870 conjugacy classes of subgroups.
| minimal degree | μ(G) = 34 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 34 points:
(1,2);(4,7)(8,10)(11,12,15,13)(14,17,18,16)(19,20)(21,24)(22,25)(23,26)(27,30)(28,31)(29,32)(33,34);(3,4,6,7)(5,8,9,10)(11,13,15,12)(14,17,18,16)(20,21)(23,31)(25,27)(29,34);(3,5,6,9)(4,8,7,10)(11,14)(12,16)(13,17)(15,18)(19,21,24,20)(22,25,30,27)(23,28,31,26)(29,32,34,33);(19,22,24,30)(20,25,21,27)(23,29,31,34)(26,32,28,33);(19,23)(20,26)(21,28)(22,29)(24,31)(25,32)(27,33)(30,34)128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7536229);