Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2883 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C42 · Schur multiplier (H2) = C29×C4 |
| direct product | C2×#20399 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×C22⋊C4 |
| commutator | C22 | C23×C42 |
| Frattini, cosocle | C22×C4 | C25 |
| prime radical | C25 | C22×C4 |
| LCS | G ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
4978 conjugacy classes of subgroups.
| minimal degree | μ(G) = 38 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 38 points:
(1,2);(3,4,6,5)(7,8)(9,18)(10,15)(11,16)(12,26)(13,17)(14,21)(19,32)(20,33)(22,28)(23,29)(24,31)(25,30)(27,35)(34,38)(36,37);(7,9,12,22,13,21,11,20)(8,14,16,28,17,18,26,33)(10,19,23,34,24,35,25,36)(15,27,30,38,31,32,29,37);(3,5,6,4)(7,10,13,24)(8,15,17,31)(9,19,21,35)(11,25,12,23)(14,27,18,32)(16,30,26,29)(20,36,22,34)(28,38,33,37);(7,11,13,12)(8,16,17,26)(9,20,21,22)(10,23,24,25)(14,28,18,33)(15,29,31,30)(19,34,35,36)(27,37,32,38)164 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6277021);