Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #45438 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
| direct product | C2×#45902 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4546 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);(3,4,7,6)(5,9,10,8)(13,19)(17,22)(20,25)(24,26)(27,28)(29,35)(30,38)(31,34)(32,37)(33,39)(36,40)(41,42);(12,16)(13,19)(14,21)(24,26)(27,28,31,34)(29,32,37,35)(30,38,39,33)(36,41,42,40);(3,5,7,10)(4,8,6,9)(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)(36,40)(38,39)(41,42);(3,6,7,4)(5,9,10,8)(11,13)(12,17)(14,20)(15,19)(16,22)(18,24)(21,25)(23,26)(27,29)(28,32)(30,36)(31,37)(33,40)(34,35)(38,41)(39,42);(11,14)(12,18)(13,20)(15,21)(16,23)(17,24)(19,25)(22,26)(27,30,31,39)(28,33,34,38)(29,36,37,42)(32,40,35,41)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7609632);