Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #53158 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
| direct product | C2×#53369 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C2×#392 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4664 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)(5,10)(6,9)(7,8)(13,19)(17,22)(20,25)(24,26)(27,28)(29,33)(30,37)(31,34)(32,35)(36,40)(38,42)(39,41);(4,8)(5,6)(12,16)(14,21)(17,22)(20,25)(27,29,32,30)(28,33,35,37)(31,36,39,38)(34,40,41,42);(3,4)(5,9)(6,10)(7,8)(11,12,15,16)(13,17,19,22)(14,18,21,23)(20,24,25,26)(27,30,32,29)(28,33,35,37)(31,38,39,36)(34,40,41,42);(3,5,7,6)(4,9,8,10)(11,13)(12,17)(14,20)(15,19)(16,22)(18,24)(21,25)(23,26)(27,31,32,39)(28,34,35,41)(29,36,30,38)(33,40,37,42);(3,6,7,5)(4,10,8,9)(11,14)(12,18)(13,20)(15,21)(16,23)(17,24)(19,25)(22,26)(27,28)(29,37)(30,33)(31,34)(32,35)(36,42)(38,40)(39,41)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8013026);