Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #20022 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
| direct product | C2×#20125 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | D42 |
| commutator, Frattini, cosocle | C42 | C25 |
| prime radical | C2×#555 | C22 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
1596 conjugacy classes of subgroups.
| minimal degree | μ(G) = 34 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 34 points:
(1,2);(3,4,8,6)(5,11,15,18)(7,16,17,10)(9,13,12,14)(19,20)(21,28)(22,26)(23,31)(24,27)(25,30)(29,34)(32,33);(3,5,8,15)(4,9,6,12)(7,13,17,14)(10,18,16,11)(21,29)(25,33)(28,34)(30,32);(3,6,8,4)(5,12,15,9)(7,16,17,10)(11,14,18,13)(19,21)(20,25)(22,30)(23,32)(24,29)(26,34)(27,33)(28,31);(5,13)(7,17)(9,18)(10,16)(11,12)(14,15)(19,22,24,23)(20,26,27,31)(21,30,29,32)(25,34,33,28)74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6266653);