Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12753 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28 |
| direct product | C2×#12755 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1790 |
| commutator | C2×D4 | C23×C4 |
| Frattini, cosocle | C2×#41 | C24 |
| prime radical | C2×#3537 | C2 |
| LCS | G ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
2799 conjugacy classes of subgroups.
| minimal degree | μ(G) = 18 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 18 points:
(1,2);(3,4,6,9,7,10,13,18)(5,8,12,16,11,15,14,17);(4,8)(6,13)(9,16)(10,15)(12,14)(17,18);(5,11)(12,14)68 conjugacy classes — view the full character table.
| maximal subgroup of | #72215601, C2×#511819, #72231756, … |
| maximal normal subgroup of | #72215601, C2×#511819, #72231756, … |
| maximal quotient of | #72215536, #72230994, #72231013, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 400761);