Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #63091 |
| automorphism | ? |
| homology | Abelianization (H1) = C27 · Schur multiplier (H2) = C220 |
| direct product | C2×#63092 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C24×D4 |
| commutator, Frattini, cosocle | C4 | C27 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×#585 ◁ G | |
| LECS | G ▷ C4 ▷ C2 ▷ C1 | |
56326 conjugacy classes of subgroups.
| minimal degree | μ(G) = 34 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 7 generators on 34 points:
(1,2);(3,4,8,13)(5,10,15,23)(6,11,17,25)(7,9,19,12)(14,22,27,32)(16,20,29,24)(18,21,30,26)(28,31,34,33);(4,9)(7,19)(10,20)(11,21)(12,13)(16,29)(18,30)(22,31)(23,24)(25,26)(28,34)(32,33);(3,5)(4,10)(6,14)(7,16)(8,15)(9,20)(11,22)(12,24)(13,23)(17,27)(18,28)(19,29)(21,31)(25,32)(26,33)(30,34);(3,6)(4,11)(5,14)(7,18)(8,17)(9,21)(10,22)(12,26)(13,25)(15,27)(16,28)(19,30)(20,31)(23,32)(24,33)(29,34);(6,17)(11,25)(14,27)(18,30)(21,26)(22,32)(28,34)(31,33);(5,15)(10,23)(14,27)(16,29)(20,24)(22,32)(28,34)(31,33)164 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10494193);