Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2668 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C25×C42 |
| direct product | C2×#14602 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C25 | C24 |
| commutator | C23 | C24×C4 |
| Frattini, cosocle | C24 | C25 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C25 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
3346 conjugacy classes of subgroups.
| minimal degree | μ(G) = 30 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 30 points:
(1,2);(3,4,6,5)(8,11)(9,13)(15,16)(17,20)(18,19)(21,22)(23,24,27,28)(25,29,30,26);(8,11)(12,14)(16,18)(17,21)(23,25,27,30)(24,26,28,29);(3,5,6,4)(7,8,10,11)(9,12,13,14)(15,17,19,21)(16,20,18,22)(23,25)(24,26)(27,30)(28,29);(7,9,10,13)(8,12,11,14)(15,18,19,16)(17,22,21,20)(23,26)(24,25)(27,29)(28,30)128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 428290);