Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #465 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C8 · Schur multiplier (H2) = C28 |
| direct product | C2×#9072 |
| subgroup | G/H | |
|---|---|---|
| center | C23×C4 | C2×D4 |
| commutator | C2×C4 | C23×C8 |
| Frattini, cosocle | C2×C42 | C24 |
| socle | C24 | C2×C22⋊C4 |
| prime radical | C22×#121 | C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C4 ◁ C23×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
2386 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);(3,4)(5,6)(8,11)(9,12)(13,14)(15,16,17,19,18,21,22,20);(7,8,12,14,10,13,9,11)(16,19,21,20);(4,6)176 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#210728, C2×#227634, C2×D4×D4⋊C8, … |
| maximal normal subgroup of | C2×#210728, C2×#227634, C2×D4×D4⋊C8, … |
| maximal quotient of | C2×#137625, C2×#137612, #72286807, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 103001);