Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #36702 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
| direct product | C2×#37077 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C25 | C24 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4432 conjugacy classes of subgroups.
| minimal degree | μ(G) = 42 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 42 points:
(1,2);(3,4,6,8)(5,7,9,10)(14,21)(18,23)(20,25)(24,26)(27,28)(29,35)(30,33)(31,34)(32,37)(36,41)(38,40)(39,42);(12,16)(13,19)(18,23)(20,25)(27,28,31,34)(29,32,37,35)(30,33,38,40)(36,39,42,41);(3,4)(5,7)(6,8)(9,10)(11,12,15,16)(13,17,19,22)(14,18,21,23)(20,24,25,26)(27,28)(29,32)(30,33)(31,34)(35,37)(36,39)(38,40)(41,42);(11,13)(12,17)(14,20)(15,19)(16,22)(18,24)(21,25)(23,26)(27,29,31,37)(28,32,34,35)(30,36,38,42)(33,39,40,41);(3,5,6,9)(4,7,8,10)(11,14)(12,18)(13,20)(15,21)(16,23)(17,24)(19,25)(22,26)(27,30,31,38)(28,33,34,40)(29,36,37,42)(32,39,35,41)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7541554);