Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #61643 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C213 |
| direct product | C2×#61796 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C26 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#532 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
6342 conjugacy classes of subgroups.
| minimal degree | μ(G) = 34 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 34 points:
(1,2);(3,4)(5,11)(6,9)(7,10)(8,13)(12,17)(14,16)(15,18)(19,20,24,29)(21,25,30,34)(22,27,32,33)(23,26,31,28);(3,5)(4,8)(6,12)(7,13)(9,15)(10,11)(14,18)(16,17)(19,21,24,30)(20,25,29,34)(22,31,32,23)(26,27,28,33);(3,6,7,14)(4,9,10,16)(5,12,13,18)(8,15,11,17)(20,26)(21,30)(23,31)(25,33)(27,34)(28,29);(5,13)(8,11)(12,18)(15,17)(19,22)(20,27)(21,31)(23,30)(24,32)(25,28)(26,34)(29,33);(4,10)(8,11)(9,16)(15,17)(21,30)(22,32)(25,34)(27,33)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10485168);