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×#53332 |
| 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 | |
4518 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)(5,8)(6,7)(9,10)(13,19)(17,22)(20,25)(24,26)(27,28,31,34)(29,32,36,39)(30,33,37,40)(35,38,41,42);(4,6)(5,9)(12,16)(14,21)(17,22)(20,25)(30,37)(33,40)(35,41)(38,42);(3,5,7,9)(4,8,6,10)(11,12,15,16)(13,17,19,22)(14,18,21,23)(20,24,25,26)(27,28)(29,32)(30,33)(31,34)(35,38)(36,39)(37,40)(41,42);(3,4,7,6)(5,8,9,10)(11,13)(12,17)(14,20)(15,19)(16,22)(18,24)(21,25)(23,26)(27,29,31,36)(28,32,34,39)(30,35,37,41)(33,38,40,42);(3,6,7,4)(5,10,9,8)(11,14)(12,18)(13,20)(15,21)(16,23)(17,24)(19,25)(22,26)(27,30)(28,33)(29,35)(31,37)(32,38)(34,40)(36,41)(39,42)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8013073);