Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #465 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C8 · Schur multiplier (H2) = C25×C4 |
| direct product | C2×#13195 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C8 | C2×D4 |
| commutator | C2×C4 | C23×C8 |
| Frattini, cosocle | C22×C8 | C24 |
| socle | C23 | C8×D4 |
| prime radical | C22×#135 | C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C8 ◁ C23×C16 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
1178 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)(7,8,13,19,14,18,28,34,11,17,21,16,12,29,10,27)(9,20,25,31,26,35,37,38,23,15,30,33,24,32,22,36);(3,4,5,6)(7,9,21,30,14,26,10,22,11,23,13,25,12,24,28,37)(8,15,16,31,18,32,27,38,17,20,19,33,29,35,34,36);(7,10,12,21,11,28,14,13)(8,16,18,27,17,19,29,34)(9,22,24,30,23,37,26,25)(15,31,32,38,20,33,35,36)176 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 402294);