Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1999 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28 |
| direct product | C2×#8313 |
| subgroup | G/H | |
|---|---|---|
| center | C23×C4 | C2×D4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| socle | C24 | C2×C22⋊C4 |
| prime radical | C22×#532 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C4 ◁ C23×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
3838 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)(7,8)(9,11)(10,13)(12,14)(15,16,18,21,19,17,22,20);(8,11)(9,14)(12,13)(15,17)(16,19)(18,20)(21,22);(4,6)(16,17)(20,21)128 conjugacy classes — view the full character table.
| maximal subgroup of | #72286602, #72286617, C2×#208698, … |
| maximal normal subgroup of | #72286602, #72286617, C2×#208698, … |
| maximal quotient of | C2×#129555, C2×#129565, #72321250, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 92045);