Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2002 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
| direct product | C4×#2006 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | #120 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C23 | C2×#120 |
| prime radical | C2×#2006 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2252 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 16 points:
(1,2,3,4);(10,14)(11,12)(15,16);(5,6)(7,8)(9,10)(11,16)(12,14)(13,15);(6,8)(9,10)(11,14)(12,16)(13,15)104 conjugacy classes — view the full character table.
| maximal subgroup of | #72282555, C4×#24222 |
| maximal normal subgroup of | #72282555, C4×#24222 |
| maximal quotient of | #72217670, #72217673, #72217677 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 266226);