Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2018 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
| direct product | C2×#11364 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | #121 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C23 | #390 |
| prime radical | C25 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1694 conjugacy classes of subgroups.
| minimal degree | μ(G) = 42 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 42 points:
(1,2);(3,4)(5,10)(6,8)(7,9)(11,12)(13,21,24,17)(14,25,28,18)(15,20)(16,19)(22,32)(23,37,29,33)(26,34)(27,38,30,35)(31,36)(39,40)(41,42);(3,5,7,6)(4,8,9,10)(11,13)(12,17)(14,26)(15,29)(16,24)(18,34)(19,21)(20,37)(22,27)(23,31)(25,42)(28,41)(30,40)(32,35)(33,36)(38,39);(3,6,7,5)(4,8,9,10)(11,14)(12,18)(13,22)(15,30)(16,28)(17,32)(19,25)(20,38)(21,39)(23,41)(24,40)(26,29)(27,31)(33,42)(34,37)(35,36)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 265783);