Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30412 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C4 |
| direct product | C2×#32040 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#2540 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C22×C4×D4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2324 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)(5,7)(6,8)(9,10)(11,12)(13,17)(14,16)(15,18)(20,25)(26,33)(27,28)(29,34);(3,5,7,4)(6,9,10,8)(11,13,14,18)(12,15,16,17)(19,20,23,25)(21,29,31,34)(22,28,24,27)(26,32,33,30);(3,6,7,10)(4,8,5,9)(12,16)(15,17)(19,21,23,31)(20,26,25,33)(22,30,24,32)(27,29,28,34);(4,5)(8,9)(13,18)(15,17)(19,22,23,24)(20,27,25,28)(21,30,31,32)(26,29,33,34)80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7414691);