Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2218 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C4 |
| direct product | C4×#2245 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C2×D8 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| socle, prime radical | C23 | C22×D8 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C43 ◁ C42×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
598 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 4 generators on 44 points:
(1,2,3,4);(5,6)(7,10)(8,9)(11,12)(13,14,20,26)(15,22,30,41)(16,21,32,39)(17,24,34,36)(18,23,27,42)(19,35,33,25)(28,40,43,38)(29,44,31,37);(6,9)(10,12)(13,15,27,43,19,29,17,16,20,30,18,28,33,31,34,32)(14,21,36,44,25,38,23,22,26,39,24,37,35,40,42,41);(5,7,8,11)(6,10,9,12)(13,16,17,31,33,43,27,30,20,32,34,29,19,28,18,15)(14,22,23,40,35,44,36,39,26,41,42,38,25,37,24,21)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 403239);