Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C82 · Schur multiplier (H2) = C26×C8 |
| direct product | C2×C8×C22⋊C8 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4×C8 | C22 |
| commutator | C2 | C22×C82 |
| Frattini, cosocle | C2×C42 | C24 |
| socle | C24 | C2×C42 |
| prime radical | C25 | C42 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
2022 conjugacy classes of subgroups.
| minimal degree | μ(G) = 22 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 22 points:
(1,2);(7,8,9,11,10,12,13,14);(3,4)(5,6)(15,16,17,19,18,20,21,22);(4,6)320 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C2×C82×D4 |
| maximal normal subgroup of | C2×C82×D4 |
| maximal quotient of | C2×#96686, C2×C8×D4⋊C8, C2×#128250, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 89078);