Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #8516 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28 |
| direct product | C2×#8520 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #379 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C23×#41 | C22 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
3388 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)(5,9,10,7)(6,8)(11,12)(13,21,24,33)(14,25,27,18)(15,19)(16,20)(17,23,31,28)(22,36)(26,40,42,32)(29,34)(30,37)(35,38)(39,41);(3,5)(4,7)(6,10)(8,9)(11,13,16,24)(12,17,20,31)(14,22,27,38)(15,28,29,23)(18,30,25,39)(19,33,34,21)(26,41,42,37)(32,36,40,35);(11,14)(12,18)(13,22)(15,26)(16,27)(17,30)(19,32)(20,25)(21,35)(23,37)(24,38)(28,41)(29,42)(31,39)(33,36)(34,40)68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 94947);