Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #47826 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C213 |
| direct product | C2×#48835 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
7820 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)(7,8,12,14)(9,11,10,13)(15,16,20,23)(17,18,21,25)(19,22,27,29)(24,26,28,30);(7,9,12,10)(8,13,14,11)(17,21)(18,25)(19,27)(22,29);(3,4,5,6)(8,14)(9,10)(15,17)(16,18)(19,24)(20,21)(22,26)(23,25)(27,28)(29,30);(7,10,12,9)(8,13,14,11)(15,18)(16,21)(17,23)(19,26)(20,25)(22,28)(24,29)(27,30);(7,11,12,13)(8,9,14,10)(15,19)(16,22)(17,24)(18,26)(20,27)(21,28)(23,29)(25,30)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7612100);