Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13429 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
| direct product | C2×#13430 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2044 |
| commutator, Frattini, cosocle | C2×#41 | C24 |
| prime radical | C2×#535 | C22 |
| LCS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
1113 conjugacy classes of subgroups.
| minimal degree | μ(G) = 34 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 34 points:
(1,2);(3,4,8,13)(5,14,15,27)(6,9,18,11)(7,12,21,28)(10,20,25,17)(16,31,30,29)(19,23,33,26)(22,34,24,32);(4,9)(5,15)(6,18)(10,22)(11,13)(12,23)(14,29)(19,33)(24,25)(26,28)(27,31)(32,34);(3,5,6,16,8,15,18,30)(4,10,11,24,13,25,9,22)(7,20,19,34,21,17,33,32)(12,27,26,31,28,14,23,29)56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 406313);