Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24805 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28×C4 |
| direct product | C2×#28500 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#121 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3244 conjugacy classes of subgroups.
| minimal degree | μ(G) = 22 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 22 points:
(1,2);(7,8,14,17,13,19,12,18)(9,10,11,21,16,22,20,15);(3,4,5,6)(7,9,12,20,13,16,14,11)(8,15,17,10,19,21,18,22);(4,6)(7,10)(8,16)(9,19)(11,17)(12,21)(13,22)(14,15)(18,20);(7,11,12,9,13,20,14,16)(8,10,17,21,19,22,18,15)80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6372786);