Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #99 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C43 · Schur multiplier (H2) = C23×C43 |
| direct product | C42×#100 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C42 | C22⋊C4 |
| commutator | C22 | C2×C43 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C23 | C22×C22⋊C4 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C42 ◁ C2×C43 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2382 conjugacy classes of subgroups; only the 870 normal subgroups are shown.
This group has 2382 conjugacy classes of subgroups — too many to draw. Only its 870 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 16 points:
(1,2,3,4);(5,6,7,8);(9,10,11,14,12,13,15,16);(10,13)(11,15)176 conjugacy classes — view the full character table.
| maximal subgroup of | C4×#21473, #72279953 |
| maximal normal subgroup of | C4×#21473, #72279953 |
| maximal quotient of | #72279558 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 260378);