Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #42996 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C213 |
| direct product | C2×#45270 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C22×#555 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
7330 conjugacy classes of subgroups.
| minimal degree | μ(G) = 30 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 30 points:
(1,2);(4,6)(7,8)(9,11)(10,13)(12,14)(15,16)(17,21)(18,22)(19,20)(23,25)(24,29)(26,28)(27,30);(7,9,12,10)(8,13,14,11)(16,19)(17,23)(18,26)(29,30);(3,4,5,6)(7,10,12,9)(8,13,14,11)(15,17,20,23)(16,21,19,25)(18,27,26,24)(22,30,28,29);(7,11,12,13)(8,9,14,10)(15,18,20,26)(16,22,19,28)(17,24,23,27)(21,29,25,30);(15,19,20,16)(17,25,23,21)(18,28,26,22)(24,30,27,29)116 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7577117);