Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1595 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
| direct product | C2×#9043 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | #120 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#121 | C22 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
1934 conjugacy classes of subgroups.
| minimal degree | μ(G) = 30 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 30 points:
(1,2);(4,6)(7,8,9,13,11,14,10,12)(15,16,19,23,20,24,18,22)(17,25,27,21,28,29,26,30);(3,4,5,6)(8,12)(9,10)(13,14)(15,17)(16,21)(18,27)(19,26)(20,28)(22,29)(23,25)(24,30);(8,13)(9,10)(12,14)(15,17)(16,21)(18,26)(19,27)(20,28)(22,25)(23,29)(24,30)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 102311);