Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #353 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C25×C42 |
| direct product | C2×#12581 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | #99 |
| commutator | C2×C4 | C22×C42 |
| Frattini, cosocle | C22×D4 | C24 |
| socle | C23 | C2×#99 |
| prime radical | C23×#41 | C22 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C23×C4 ◁ C23×#41 ◁ G | |
| LECS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
2658 conjugacy classes of subgroups.
| minimal degree | μ(G) = 22 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 22 points:
(1,2);(7,8)(9,13)(10,14,18,12)(11,15)(16,20,22,19)(17,21);(3,4,5,6)(8,12,15,14)(13,19,21,20);(7,9,11,17)(8,13,15,21)(10,16,18,22)(12,19,14,20)104 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#146667, C2×#147577, C2×#146656, … |
| maximal normal subgroup of | C2×#146667, C2×#147577, C2×#146656, … |
| maximal quotient of | C2×#128057, #72240342, D3×#12581, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 400092);