Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1510 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C8 · Schur multiplier (H2) = C26 |
| direct product | C2×#11469 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C2×C22⋊C4 |
| commutator | C2×C4 | C23×C8 |
| Frattini, cosocle | C22×C8 | C24 |
| socle | C23 | C2×C22⋊C8 |
| prime radical | C2×#525 | C4 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C24×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
1056 conjugacy classes of subgroups.
| minimal degree | μ(G) = 42 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 42 points:
(1,2);(3,4)(5,9,10,7)(6,8)(11,12,16,20,17,22,29,38,18,23,14,28,15,21,30,37)(13,24,25,41,26,36,42,34,27,39,32,40,31,19,33,35);(3,5)(4,7)(6,10)(8,9)(11,13)(12,19)(14,25)(15,31)(16,32)(17,26)(18,27)(20,34)(21,39)(22,24)(23,36)(28,35)(29,33)(30,42)(37,41)(38,40);(11,14,17,30,18,16,15,29)(12,20,22,38,23,28,21,37)(13,25,26,42,27,32,31,33)(19,34,24,40,36,35,39,41)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 387277);