Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #10631 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
| direct product | C2×#10636 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #397 |
| commutator, Frattini, cosocle | C2×C42 | C24 |
| prime radical | C24 | #124 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
676 conjugacy classes of subgroups.
| minimal degree | μ(G) = 34 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 34 points:
(1,2);(3,4,7,9,8,12,6,11)(5,13,15,18,14,17,16,10)(19,20)(21,24)(22,25)(23,26)(27,34)(28,32)(29,31)(30,33);(4,9)(5,14)(6,7)(10,17)(11,12)(13,18)(19,21,22,28)(20,24,25,32)(23,27,30,29)(26,34,33,31);(3,5,6,15,8,14,7,16)(4,10,11,13,12,18,9,17)(20,25)(21,27)(23,30)(24,31)(28,29)(32,34)68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 150154);