Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #120 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C8 · Schur multiplier (H2) = C25×C4 |
| direct product | C2×#12592 |
| subgroup | G/H | |
|---|---|---|
| center | C23×C8 | C23 |
| commutator | C22 | C22×C4×C8 |
| Frattini, cosocle | C22×C8 | C24 |
| socle, prime radical | C24 | C22×C8 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
866 conjugacy classes of subgroups.
| minimal degree | μ(G) = 46 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 4 generators on 46 points:
(1,2);(7,8)(9,12)(10,11)(13,14)(15,16,18,22,19,23,28,34,20,24,29,35,30,36,40,44)(17,21,26,32,27,33,39,43,25,31,37,41,38,42,45,46);(3,4,5,6)(8,11)(12,14)(15,17)(16,21)(18,26)(19,27)(20,25)(22,32)(23,33)(24,31)(28,39)(29,37)(30,38)(34,43)(35,41)(36,42)(40,45)(44,46);(7,9,10,13)(8,12,11,14)(17,25)(21,31)(26,37)(27,38)(32,41)(33,42)(39,45)(43,46)224 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 400110);