Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C8 · Schur multiplier (H2) = C28 |
| direct product | C22×#1277 |
| subgroup | G/H | |
|---|---|---|
| center | C24×C4 | D4 |
| commutator | C4 | C22×C4×C8 |
| Frattini, cosocle | C2×C42 | C24 |
| socle, prime radical | C25 | C4⋊C4 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24×C4 ◁ C23×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
2490 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2);(3,4);(5,6)(7,8)(10,14)(11,12)(15,16)(17,18,19,21,20,22,23,24);(6,8)(9,10,12,15,13,16,11,14)224 conjugacy classes — view the full character table.
| maximal subgroup of | C22×#8200, C22×#8947, C22×#8830, … |
| maximal normal subgroup of | C22×#8200, C22×#8947, C22×#8830, … |
| maximal quotient of | C22×#7156 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 88104);