Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #392 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C212 |
| direct product | C2×C4×#392 |
| subgroup | G/H | |
|---|---|---|
| center | C24×C4 | C23 |
| commutator | C23 | C24×C4 |
| Frattini, cosocle | C24 | C25 |
| socle | C25 | C24 |
| prime radical | C22×#392 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
13386 conjugacy classes of subgroups.
| minimal degree | μ(G) = 18 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 18 points:
(1,2);(3,4,5,6);(7,8)(9,10)(11,12)(13,14);(8,10)(15,16)(17,18);(12,14)(16,18)176 conjugacy classes — view the full character table.
| maximal subgroup of | C2×C4×#2397, C2×C4×#392⋊C3, #72217661, … |
| maximal normal subgroup of | C2×C4×#2397, C2×C4×#392⋊C3, #72217661, … |
| maximal quotient of | C2×#359042, C2×C4×#2017, C2×C4×#2010, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 421223);