Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of D16 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C8 · Schur multiplier (H2) = C27 |
| direct product | C22×#1331 |
| subgroup | G/H | |
|---|---|---|
| center | C23×C4 | D8 |
| commutator | C8 | C23×C8 |
| Frattini, cosocle | C4×C8 | C24 |
| socle | C24 | #102 |
| prime radical | C23×D8 | C4 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C4 ◁ C22×C42 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
1310 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 28 points:
(1,2);(3,4);(5,6,7,9,8,10,11,12)(13,14)(15,18)(16,25)(17,21)(19,22)(20,23)(24,27)(26,28);(14,18)(15,22)(16,23)(19,20)(21,27)(24,28)(25,26)176 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#128531, C22×C8×D16, C2×C8×#357, … |
| maximal normal subgroup of | C2×#128531, C22×C8×D16, C2×C8×#357, … |
| maximal quotient of | C2×#150058, C2×#150064, C2×#150057, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 388246);