Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #465 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C8 · Schur multiplier (H2) = C25 |
| direct product | C2×#11996 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C8 | C2×D4 |
| commutator | C2×C4 | C23×C8 |
| Frattini, cosocle | C4×C8 | C24 |
| socle | C23 | C2×C4⋊C8 |
| prime radical | C24 | C4⋊C8 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C8 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
638 conjugacy classes of subgroups.
| minimal degree | μ(G) = 26 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 26 points:
(1,2);(4,7)(5,9)(8,10)(11,12,13,17,14,18,19,24,15,16,20,22,21,23,25,26);(3,4,5,7,6,8,9,10);(12,16)(17,22)(18,23)(24,26)176 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#135732, C2×#122921, C2×#135714, … |
| maximal normal subgroup of | C2×#135732, C2×#122921, C2×#135714, … |
| maximal quotient of | C2×#122921, #72231926, #72231931 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 390512);