Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #52308 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
| direct product | C2×#52805 |
| 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 | |
4534 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)(5,9)(6,8)(7,10)(12,16)(13,19)(14,22)(24,26)(27,28)(29,30)(31,35)(32,33)(34,37)(36,38)(39,40)(41,42);(3,5,8,7)(4,9,6,10)(28,33)(30,34)(35,40)(38,41);(3,6,8,4)(5,9,7,10)(11,12,15,16)(13,20,19,17)(14,18,22,23)(21,26,25,24)(27,29)(28,34)(30,33)(31,36)(32,37)(35,41)(38,40)(39,42);(3,7,8,5)(4,9,6,10)(11,13,15,19)(12,17,16,20)(14,21,22,25)(18,24,23,26)(27,30,32,34)(28,29,33,37)(31,38,39,41)(35,36,40,42);(11,14)(12,18)(13,21)(15,22)(16,23)(17,24)(19,25)(20,26)(27,31,32,39)(28,35,33,40)(29,36,37,42)(30,38,34,41)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7999842);