Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3026 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C210 |
| direct product | C2×#21484 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#120 |
| commutator | C23 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| socle | C22 | C4×#120 |
| prime radical | C2×#3025 | C2 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
4494 conjugacy classes of subgroups.
| minimal degree | μ(G) = 34 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 34 points:
(1,2);(3,4,8,7,9,12,21,20)(5,10,16,14,15,23,30,29)(6,11,19,17,18,24,32,31)(13,25,28,34,27,22,33,26);(3,5)(4,10)(6,13)(7,14)(8,16)(9,15)(11,22)(12,23)(17,26)(18,27)(19,28)(20,29)(21,30)(24,25)(31,34)(32,33);(3,6)(4,11)(5,13)(7,17)(8,19)(9,18)(10,22)(12,24)(14,26)(15,27)(16,28)(20,31)(21,32)(23,25)(29,34)(30,33);(3,7,8,12,9,20,21,4)(5,14,16,23,15,29,30,10)(6,17,19,24,18,31,32,11)(13,26,28,25,27,34,33,22)116 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6279937);