Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #53390 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
| direct product | C2×#57651 |
| 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 | |
4572 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);(4,7)(5,9)(11,12)(13,19)(14,22)(15,21)(16,17)(18,23)(20,25)(24,26)(27,28,32,35)(29,33,30,37)(31,34,39,42)(36,40,38,41);(4,7)(8,10)(12,17)(14,23)(15,19)(20,24)(27,29,32,30)(28,33,35,37)(31,38,39,36)(34,41,42,40);(3,4,6,7)(5,8,9,10)(11,13)(12,15)(14,24)(16,21)(17,19)(18,26)(20,23)(22,25)(27,28)(29,33)(30,37)(31,34)(32,35)(36,40)(38,41)(39,42);(3,5,6,9)(4,8,7,10)(11,14)(12,18)(13,20)(15,25)(16,23)(17,22)(19,26)(21,24)(27,30,32,29)(28,33,35,37)(31,38,39,36)(34,40,42,41);(11,15)(12,13)(14,25)(16,19)(17,21)(18,20)(22,24)(23,26)(27,31)(28,34)(29,36)(30,38)(32,39)(33,40)(35,42)(37,41)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8008612);