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×#18377 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#120 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#3490 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
3564 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);(3,4,7,12)(5,14,15,27)(6,18,21,11)(8,13,22,10)(9,19,24,16)(17,29,30,34)(20,26,33,28)(23,32,25,31);(3,5)(4,9)(6,19)(7,15)(8,17)(10,25)(11,27)(12,24)(13,23)(14,18)(16,21)(20,32)(22,30)(26,29)(28,34)(31,33);(6,20)(11,28)(14,29)(16,31)(18,26)(19,32)(21,33)(27,34);(4,10)(5,15)(9,23)(11,26)(12,13)(14,29)(16,19)(17,30)(18,28)(24,25)(27,34)(31,32)86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6254147);