Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #53390 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
| direct product | C2×#56387 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
5586 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);(3,4)(5,7)(6,8)(9,10)(19,20,24,29)(21,27,32,25)(22,26,28,23)(30,31,33,34);(6,10)(8,9)(11,12)(13,17)(14,16)(15,18)(19,21,24,32)(20,25,29,27)(22,33,28,30)(23,34,26,31);(4,5)(8,9)(11,12,14,16)(13,15,18,17)(19,22,24,28)(20,26,29,23)(21,30,32,33)(25,34,27,31);(3,5,7,4)(6,9,10,8)(11,12)(13,15)(14,16)(17,18)(19,21)(20,27)(22,33)(23,31)(24,32)(25,29)(26,34)(28,30);(3,6)(4,8)(5,9)(7,10)(11,13)(12,15)(14,18)(16,17)(19,23,24,26)(20,28,29,22)(21,31,32,34)(25,33,27,30)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8005992);