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×#55727 |
| 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 | |
4648 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)(14,16)(15,22)(17,20)(18,23)(21,24)(25,26)(27,28,32,35)(29,33,30,37)(31,34,39,42)(36,40,38,41);(4,7)(8,10)(12,14)(13,20)(18,22)(21,26)(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)(12,17)(14,19)(15,21)(16,20)(18,24)(22,25)(23,26)(27,30,32,29)(28,33,35,37)(31,38,39,36)(34,40,42,41);(3,5)(4,8)(6,9)(7,10)(11,14,16,12)(13,19,20,17)(15,22,23,18)(21,25,26,24)(27,28)(29,33)(30,37)(31,34)(32,35)(36,40)(38,41)(39,42);(11,15,16,23)(12,18,14,22)(13,21,20,26)(17,24,19,25)(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, 8007054);