Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2230 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
| direct product | C2×#12824 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #476 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| prime radical | C22×#135 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
1114 conjugacy classes of subgroups.
| minimal degree | μ(G) = 38 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 38 points:
(1,2);(4,6)(7,8)(9,19)(10,23,25,30)(11,27)(12,14)(13,18)(15,21,33,37)(16,20)(17,22)(24,36,26,35)(28,32,38,34)(29,31);(3,4,5,6)(7,9,20,29,13,22,11,12)(8,14,27,17,18,31,16,19)(10,21,26,38,25,37,24,28)(15,30,34,35,33,23,32,36);(7,10)(8,15)(9,21)(11,24)(12,28)(13,25)(14,30)(16,32)(17,35)(18,33)(19,36)(20,26)(22,37)(23,31)(27,34)(29,38)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 400983);