Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3264 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C28 |
| direct product | C4×#3350 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22×D4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| socle | C23 | C23×D4 |
| prime radical | C2×#553 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C42×D4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1970 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2,3,4);(5,6,9,8)(7,11,12,10)(13,14)(15,22)(16,20)(17,25)(18,21)(19,24)(23,28)(26,27);(5,7)(6,10)(8,11)(9,12)(15,23)(19,27)(22,28)(24,26);(5,8,9,6)(7,11,12,10)(13,15,18,23)(14,19,21,27)(16,24,17,26)(20,28,25,22);(7,12)(10,11)(13,16,18,17)(14,20,21,25)(15,24,23,26)(19,28,27,22)116 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6291988);