Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #45438 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C213 |
| direct product | C2×#45490 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C22×#560 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
7734 conjugacy classes of subgroups.
| minimal degree | μ(G) = 30 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 30 points:
(1,2);(4,6)(9,13)(12,14)(16,20)(17,23)(18,25)(28,30);(8,11)(9,13)(15,16,19,20)(17,21,23,27)(18,26,25,22)(24,30,29,28);(3,4,5,6)(7,8)(9,12)(10,11)(13,14)(15,16)(17,21)(18,22)(19,20)(23,27)(24,28)(25,26)(29,30);(7,9)(8,12)(10,13)(11,14)(15,17,19,23)(16,21,20,27)(18,24,25,29)(22,28,26,30);(7,8)(9,12)(10,11)(13,14)(15,18,19,25)(16,22,20,26)(17,24,23,29)(21,28,27,30)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7609433);