Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33856 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
| direct product | C2×#33856 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | D4×D8 |
| commutator, Frattini, cosocle | C2×C8 | C25 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×#443 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
1928 conjugacy classes of subgroups.
| minimal degree | μ(G) = 34 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 34 points:
(1,2);(4,10)(5,15)(6,19)(7,8)(11,12)(13,29)(14,28)(16,32)(17,33)(18,24)(20,25)(22,23)(26,27)(30,31);(3,4)(5,16)(6,20)(7,13)(8,25)(9,14)(10,19)(11,18)(12,17)(15,30)(21,31)(22,28)(23,29)(24,27)(26,34)(32,33);(3,5)(4,11)(6,17)(7,24)(8,15)(9,18)(10,26)(12,29)(13,30)(14,16)(19,21)(20,32)(22,34)(23,33)(25,27)(28,31);(3,5)(4,12)(6,21)(7,15)(8,24)(9,18)(10,27)(11,29)(13,31)(14,32)(16,20)(17,19)(22,33)(23,34)(25,26)(28,30)86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7530852);