Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3057 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C210 |
| direct product | C2×#20669 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22×D4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| socle | C23 | C22×C4⋊C4 |
| prime radical | C23×#41 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C23×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
4594 conjugacy classes of subgroups.
| minimal degree | μ(G) = 38 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 38 points:
(1,2);(3,4)(5,6)(7,8,12,16,13,17,11,26)(9,18,22,28,20,32,21,14)(10,15,25,30,23,29,24,36)(19,31,34,37,35,38,33,27);(7,9,11,21,13,20,12,22)(8,14,16,18,17,28,26,32)(10,19,24,33,23,35,25,34)(15,27,30,31,29,37,36,38);(4,6)(7,10)(8,15)(9,19)(11,24)(12,25)(13,23)(14,27)(16,30)(17,29)(18,31)(20,35)(21,33)(22,34)(26,36)(28,37)(32,38);(9,20)(10,23)(14,28)(15,29)(18,32)(21,22)(24,25)(30,36)128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6278017);