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) = C25 |
| direct product | C22×#1398 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C8 | C42 |
| commutator | C4 | C22×C4×C8 |
| Frattini, cosocle | C2×C16 | C24 |
| socle | C23 | #346 |
| prime radical | C24 | M32 |
| LCS | G ▷ C4 ▷ C1 | |
| UCS | C1 ◁ C22×C8 ◁ G | |
| LECS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
522 conjugacy classes of subgroups; only the 442 normal subgroups are shown.
This group has 522 conjugacy classes of subgroups — too many to draw. Only its 442 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2);(3,4);(5,6,7,12,8,14,17,27,9,15,18,25,19,29,31,35,10,13,16,26,20,28,30,36,21,11,22,23,32,24,33,34);(6,11,13,15)(7,16)(12,25,26,23)(14,24,28,29)(17,30)(18,22)(27,35,36,34)(31,33)176 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 419742);