Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #360 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C8 · Schur multiplier (H2) = C27 |
| direct product | C22×#1333 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | #102 |
| commutator | C8 | C23×C8 |
| Frattini, cosocle | C4×C8 | C24 |
| prime radical | C23×D8 | C4 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C22×C42 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
1290 conjugacy classes of subgroups; only the 402 normal subgroups are shown.
This group has 1290 conjugacy classes of subgroups — too many to draw. Only its 402 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) = 256 |
A minimal faithful permutation representation — 4 generators on 28 points:
(1,2);(3,4);(5,6,7,9,8,10,11,12)(13,14,16,20,17,22,27,21)(15,23,25,19,26,18,24,28);(14,18)(15,24)(16,27)(19,20)(21,28)(22,23)(25,26)128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 388255);