Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3532 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C4 · Schur multiplier (H2) = C213 |
| direct product | C2×#60254 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C25 |
| commutator | C22 | C25×C4 |
| Frattini, cosocle | C2×C4 | C26 |
| socle | C23 | C24×C4 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
6080 conjugacy classes of subgroups.
| minimal degree | μ(G) = 34 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 34 points:
(1,2);(3,4)(5,11)(6,9)(7,10)(8,13)(12,17)(14,16)(15,18)(19,20,22,26,23,24,29,31)(21,25,28,32,27,30,33,34);(3,5,7,13)(4,8,10,11)(6,14,15,12)(9,17,18,16);(3,6,7,15)(4,9,10,18)(5,12,13,14)(8,16,11,17)(20,24)(25,30)(26,31)(32,34);(4,10)(8,11)(9,18)(16,17)(19,21)(20,25)(22,28)(23,27)(24,30)(26,32)(29,33)(31,34);(21,27)(25,30)(28,33)(32,34)152 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10481524);