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×#54620 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
6122 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)(11,12)(13,15)(14,17)(16,18)(19,20,24,25)(21,22,29,26)(23,31,32,27)(28,33,34,30);(6,10)(8,9)(12,13)(16,17)(20,25)(22,26)(23,32)(28,34);(4,5)(8,9)(11,12,15,13)(14,16,18,17)(19,21,24,29)(20,26,25,22)(23,28,32,34)(27,33,31,30);(3,5,7,4)(6,9,10,8)(11,13,15,12)(14,17,18,16)(19,22,24,26)(20,21,25,29)(23,30,32,33)(27,28,31,34);(3,6,7,10)(4,8,5,9)(11,14)(12,16)(13,17)(15,18)(19,23)(20,27)(21,28)(22,30)(24,32)(25,31)(26,33)(29,34)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8006000);