Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #8479 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
| direct product | C2×#8484 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #379 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| prime radical | C2×#3561 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
2270 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);(4,7,8,9)(11,12,15,20,16,19,30,34)(13,21,24,35,23,17,28,32)(14,25,27,36,26,18,29,33)(22,37,39,42,38,31,40,41);(3,4)(5,9)(6,8)(7,10)(11,13)(12,17)(14,22)(15,28)(16,23)(18,31)(19,21)(20,35)(24,30)(25,37)(26,38)(27,40)(29,39)(32,34)(33,42)(36,41);(5,10)(7,9)(11,14,16,26)(12,18,19,25)(13,22,23,38)(15,29,30,27)(17,31,21,37)(20,36,34,33)(24,40,28,39)(32,42,35,41)74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 94419);