Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3532 |
| automorphism | ? |
| homology | Abelianization (H1) = C27 · Schur multiplier (H2) = C218×C4 |
| direct product | C22×#3549 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C22 | C27 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C22 ▷ C1 | |
42046 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 7 generators on 20 points:
(1,2);(3,4);(6,8)(10,11)(14,17)(15,16);(5,6,9,8)(7,11,12,10)(13,14,18,17)(15,20,16,19);(13,15,18,16)(14,19,17,20);(5,7,9,12)(6,10,8,11)(13,16,18,15)(14,20,17,19);(5,8,9,6)(7,11,12,10)(13,17,18,14)(15,20,16,19)152 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10493353);