Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #18746 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
| direct product | C2×#18753 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#120 |
| commutator, Frattini, cosocle | C42 | C25 |
| prime radical | C23×D4 | D4 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
3160 conjugacy classes of subgroups.
| minimal degree | μ(G) = 34 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 34 points:
(1,2);(3,4,8,11)(5,12,14,17)(6,10,7,15)(9,16,18,13)(19,20)(21,27)(22,30)(23,25)(24,29)(26,31)(28,34)(32,33);(3,5,8,14)(4,9,11,18)(6,13,7,16)(10,17,15,12)(22,31)(26,30)(27,34)(29,32);(3,6,8,7)(4,10,11,15)(5,13,14,16)(9,17,18,12)(19,21)(20,24)(22,32)(23,28)(25,33)(26,34)(27,30)(29,31);(20,25)(24,33)(26,30)(27,34)86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6255862);