Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of D16 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C8 · Schur multiplier (H2) = C23×C4 |
| direct product | C8×#357 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C8 | D8 |
| commutator | C8 | C2×C4×C8 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C23 | C4×D8 |
| prime radical | C22×D8 | C2×C4 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C8 ◁ C2×C4×C8 ◁ C2×C82 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
414 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 3 generators on 28 points:
(5,6,7,9,8,10,11,12);(1,2,3,4)(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 | C8×#2177, C2×C8×#357, #72225384, … |
| maximal normal subgroup of | C8×#2177, C2×C8×#357, #72225384, … |
| maximal quotient of | C2×C8×#357, C8×#1419, C8×#1418, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 35860);