Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2184 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24 |
| direct product | C2×#12076 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C2×D8 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C4×C8 | C24 |
| socle, prime radical | C23 | C4×D8 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C2×C4×C8 ◁ C2×C82 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
474 conjugacy classes of subgroups.
| minimal degree | μ(G) = 42 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 4 generators on 42 points:
(1,2);(3,4,6,5,7,8,10,9)(11,12,15,19)(13,21,22,40)(14,25,28,39)(16,30,27,20)(17,32,33,23)(18,35,37,29)(24,41,31,36)(26,38,42,34);(11,13,15,22)(12,17,19,33)(14,26,28,42)(16,31,27,24)(18,36,37,41)(20,38,30,34)(21,39,40,25)(23,35,32,29);(3,5,6,8,7,9,10,4)(13,22)(17,33)(21,40)(23,32)(24,31)(26,42)(34,38)(36,41)116 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 390976);