Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #18376 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
| direct product | C2×#18383 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#120 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C23×D4 | D4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
3188 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);(3,4,8,13)(5,15,19,31)(6,12,7,22)(9,14,25,32)(10,16,28,17)(11,21,23,24)(18,26,20,30)(27,33,29,34);(3,5,8,19)(4,10,13,28)(6,17,7,16)(9,20,25,18)(11,30,23,26)(12,31,22,15)(14,29,32,27)(21,33,24,34);(3,4)(5,10)(6,11)(7,23)(8,13)(9,14)(12,24)(15,33)(16,26)(17,30)(18,27)(19,28)(20,29)(21,22)(25,32)(31,34);(3,6,8,7)(4,11,13,23)(5,16,19,17)(9,21,25,24)(10,26,28,30)(12,14,22,32)(15,29,31,27)(18,33,20,34)86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6254153);