Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #1343 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C24 |
| direct product | C2×#7105 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | #99 |
| commutator | C22×C4 | C22×C8 |
| Frattini, cosocle | C2×#97 | C23 |
| socle | C23 | #323 |
| prime radical | C23×D4 | C8 |
| LCS | G ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C23×C4 ◁ C22×#97 ◁ G | |
| LECS | G ▷ C2×#97 ▷ C22×C4 ▷ C2 ▷ C1 | |
686 conjugacy classes of subgroups; only the 146 normal subgroups are shown.
This group has 686 conjugacy classes of subgroups — too many to draw. Only its 146 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) = 128 |
A minimal faithful permutation representation — 3 generators on 26 points:
(1,2);(3,4)(5,7)(6,9)(8,10)(11,12,13,17,14,18,20,24,15,16,19,22,21,23,25,26);(4,7)(5,10)(8,9)(12,16)(18,23)92 conjugacy classes — view the full character table.
| maximal subgroup of | #72213167, #72213261, C22×#7105, … |
| maximal normal subgroup of | #72213167, #72213261, C22×#7105, … |
| maximal quotient of | C2×#92827, #72213553, C22×#7105 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 31249);