Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8 |
| automorphism | ? |
| homology | Abelianization (H1) = C42×C8 · Schur multiplier (H2) = C22×C42 |
| direct product | C4×#1376 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C8 | D4 |
| commutator | C4 | C42×C8 |
| Frattini, cosocle | C23×C8 | C23 |
| socle | C23 | C2×C4⋊C8 |
| prime radical | C24 | C2×M16 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C8 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
866 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2,3,4);(5,6)(7,8)(9,10,12,17,13,16,20,23,14,18,21,24,11,15,19,22);(6,8)(10,15,18,16)(17,22,24,23)224 conjugacy classes — view the full character table.
| maximal subgroup of | C4×#12607, #72224845, #72224846, … |
| maximal normal subgroup of | C4×#12607, #72224845, #72224846, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 53495);