Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11502 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
| direct product | C2×#11503 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#99 |
| commutator | C24 | C23×C4 |
| Frattini, cosocle | C2×C22⋊C4 | C24 |
| prime radical | C24×D4 | C4 |
| LCS | G ▷ C24 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ C22×#117 ◁ G | |
| LECS | G ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
2194 conjugacy classes of subgroups; only the 234 normal subgroups are shown.
This group has 2194 conjugacy classes of subgroups — too many to draw. Only its 234 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 26 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 26 points:
(1,2);(3,4,5,8,6,7,9,10)(11,12)(13,17)(14,21,22,16)(15,18)(19,25)(20,24,26,23);(12,16)(13,19)(17,24)(18,21)(20,26)(23,25);(4,7)(8,10)(11,13,15,19)(12,17,18,25)(14,20,22,26)(16,23,21,24)68 conjugacy classes — view the full character table.
| maximal subgroup of | #72318369, C2×#134094, #72318895, … |
| maximal normal subgroup of | #72318369, C2×#134094, #72318895, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 387443);