Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3264 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C28 |
| direct product | C2×#18025 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22×D4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C42 | C25 |
| socle | C23 | C2×C4×D4 |
| prime radical | C2×#3106 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C2×#431 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
1792 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)(5,11)(6,15)(7,12)(8,17)(9,13)(10,14)(16,18)(19,20,22,25,23,26,29,32)(21,24,28,31,27,30,33,34);(3,5,9,6)(4,10,13,18)(7,14,8,16)(11,12,15,17);(3,6,9,5)(4,11,13,15)(7,14,8,16)(10,12,18,17)(19,21,22,28,23,27,29,33)(20,24,25,31,26,30,32,34);(3,7,9,8)(4,12,13,17)(5,14,6,16)(10,15,18,11)(21,27)(24,30)(28,33)(31,34)116 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6252124);