Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(3,ℤ/4) |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C8 · Schur multiplier (H2) = C26 |
| direct product | C22×#1313 |
| subgroup | G/H | |
|---|---|---|
| center | C23×C4 | C42 |
| commutator | C4 | C22×C4×C8 |
| Frattini, cosocle | C4×C8 | C24 |
| socle, prime radical | C24 | #97 |
| LCS | G ▷ C4 ▷ C1 | |
| UCS | C1 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
722 conjugacy classes of subgroups; only the 562 normal subgroups are shown.
This group has 722 conjugacy classes of subgroups — too many to draw. Only its 562 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 4 generators on 28 points:
(1,2);(3,4);(13,14,15,19,16,21,23,27,17,20,22,28,24,18,25,26);(5,6,7,9,8,10,11,12)(14,18,20,21)(15,22)(19,27,28,26)(23,25)176 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 387100);