Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2883 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C8 · Schur multiplier (H2) = C210 |
| direct product | C2×#33337 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C22×C22⋊C4 |
| commutator | C22 | C24×C8 |
| Frattini, cosocle | C2×C8 | C25 |
| socle | C22 | C22×C22⋊C8 |
| prime radical | C23×#41 | C4 |
| LCS | G ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×#131 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
3978 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);(3,4,6,11,7,12,17,24,8,9,16,20,18,21,28,31)(5,10,14,22,15,23,27,32,13,19,25,29,26,30,33,34);(4,9)(6,16)(10,19)(12,21)(14,25)(17,28)(23,30)(27,33);(3,5)(4,10)(6,14)(7,15)(8,13)(9,19)(11,22)(12,23)(16,25)(17,27)(18,26)(20,29)(21,30)(24,32)(28,33)(31,34);(5,13)(10,19)(14,25)(15,26)(22,29)(23,30)(27,33)(32,34)164 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7529629);