Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2218 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
| direct product | C2×#12086 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C2×D8 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C4×C8 | C24 |
| socle | C23 | C4×D8 |
| prime radical | C22×D16 | C4 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C2×C4×C8 ◁ C2×C82 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
538 conjugacy classes of subgroups; only the 202 normal subgroups are shown.
This group has 538 conjugacy classes of subgroups — too many to draw. Only its 202 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) = 256 |
A minimal faithful permutation representation — 4 generators on 26 points:
(1,2);(3,4,7,5,6,8,10,9)(11,12,14,17)(13,16,22,24)(15,23,21,19)(18,26,25,20);(4,8)(5,9)(12,16)(13,20)(15,21)(17,24)(18,19)(22,26)(23,25);(3,5,7,8,6,9,10,4)(12,17)(16,24)(18,25)(19,23)104 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#483943, C2×#151129, #72260357, … |
| maximal normal subgroup of | C2×#483943, C2×#151129, #72260357, … |
| maximal quotient of | C2×#151129, #72232021, #72232022, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 391018);