Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #99 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C8 · Schur multiplier (H2) = C25×C4 |
| direct product | C2×C4×#349 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C42 | C22⋊C4 |
| commutator | C22 | C22×C4×C8 |
| Frattini, cosocle | C22×C8 | C24 |
| socle | C23 | C2×C22⋊C8 |
| prime radical | C25 | C2×C8 |
| LCS | G ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C42 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
1458 conjugacy classes of subgroups.
| minimal degree | μ(G) = 22 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 22 points:
(1,2);(3,4,5,6);(7,8,9,13,10,14,16,20,11,12,15,18,17,19,21,22);(8,12)(9,15)(14,19)(16,21)176 conjugacy classes — view the full character table.
| maximal subgroup of | C4×#12607, #72224843, #72224844, … |
| maximal normal subgroup of | C4×#12607, #72224843, #72224844, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 400135);