Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #453 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C29 |
| direct product | C2×#10749 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | #120 |
| commutator | C2×C4 | C22×C42 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C24×D4 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
5450 conjugacy classes of subgroups.
| minimal degree | μ(G) = 18 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 18 points:
(1,2);(3,4,5,6)(12,15)(13,17)(16,18);(7,8,9,10)(11,12)(13,18)(14,16)(15,17);(13,17)(15,18)128 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#147779, D3×#10749, C2×#146601 |
| maximal normal subgroup of | C2×#147779, C2×#146601 |
| maximal quotient of | C2×#107130, D3×#10749, C2×#106744 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 260631);