Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24264 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
| direct product | C2×#28950 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×C4×C4⋊C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1870 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);(3,4,7,9)(5,10,6,8)(11,12)(13,16)(14,15)(17,18)(19,20,22,25)(21,27,29,24)(23,26,31,33)(28,32,30,34);(3,5,7,6)(4,8,9,10)(19,21,23,30,22,29,31,28)(20,24,26,32,25,27,33,34);(3,6,7,5)(4,8,9,10)(12,15)(16,18)(20,25)(21,28)(23,31)(24,32)(27,34)(29,30);(4,9)(8,10)(11,13,14,17)(12,16,15,18)(20,25)(24,27)(26,33)(32,34)116 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6367382);