Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #36702 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C213 |
| direct product | C2×#36745 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
7814 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)(15,16)(17,18)(19,23)(20,21)(22,26)(24,25)(27,28)(29,30);(8,11)(12,14)(16,21)(17,22)(23,28)(24,29);(3,4,5,6)(7,8)(9,12)(10,11)(13,14)(15,17,20,22)(16,18,21,26)(19,24,27,29)(23,25,28,30);(7,8)(9,12)(10,11)(13,14)(15,18)(16,22)(17,21)(19,25)(20,26)(23,29)(24,28)(27,30);(7,9)(8,12)(10,13)(11,14)(15,19,20,27)(16,23,21,28)(17,24,22,29)(18,25,26,30)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7541378);