Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3003 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C214 |
| direct product | C22×#3109 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator, Frattini, cosocle | C2×C4 | C26 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C23×C42 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
10946 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 20 points:
(1,2);(3,4);(5,6)(7,10)(8,9)(11,12)(13,14)(15,20)(16,18)(17,19);(6,9)(10,12)(14,18)(19,20);(14,19)(15,17)(18,20);(5,7,8,11)(6,10,9,12)(13,15,16,17)(14,20,18,19)128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10482267);