Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #8603 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
| direct product | C2×#11260 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | #117 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C23 | #382 |
| prime radical | C25 | #41 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C22×#117 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1654 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);(7,8,12,18,13,16,11,17)(9,19,21,31,22,14,26,29)(10,23,24,30,25,32,27,15)(20,34,36,33,37,38,35,28);(3,4)(5,6)(7,9,11,21,13,22,12,26)(8,14,17,29,16,19,18,31)(10,20,24,35,25,37,27,36)(15,28,23,38,30,33,32,34);(4,6)(7,10)(8,15,16,30)(9,20)(11,24)(12,27)(13,25)(14,28,19,33)(17,23,18,32)(21,35)(22,37)(26,36)(29,38,31,34)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 265465);