Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #18992 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
| direct product | C2×#19009 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#120 |
| commutator, Frattini, cosocle | C42 | C25 |
| prime radical | C2×#3531 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
2700 conjugacy classes of subgroups.
| minimal degree | μ(G) = 34 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 34 points:
(1,2);(4,9)(5,13)(6,7)(11,12)(14,15)(16,17)(19,20,24,21)(22,29,30,34)(23,28,33,26)(25,32,27,31);(3,4)(5,10)(6,11)(7,9)(8,12)(13,17)(14,18)(15,16)(19,21,24,20)(22,27,30,25)(23,26,33,28)(29,32,34,31);(3,5)(4,10)(6,13)(7,15)(8,14)(9,16)(11,17)(12,18)(19,22)(20,25)(21,27)(23,32)(24,30)(26,34)(28,29)(31,33);(3,6,8,7)(4,11,12,9)(5,13,14,15)(10,17,18,16)68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6258588);