Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1655 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
| direct product | C2×#8919 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | #120 |
| commutator | C42 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| socle | C23 | #385 |
| prime radical | C24 | #123 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
850 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);(3,4,7,12,9,13,8,11)(5,14,10,17,16,18,15,6)(20,24)(22,29)(25,33)(26,30)(27,31)(32,34);(3,5,7,10,9,16,8,15)(4,6,12,14,13,17,11,18)(19,20)(21,25)(22,30)(23,26)(24,29)(27,32)(28,34)(31,33);(3,6,9,17)(4,10,13,15)(5,12,16,11)(7,14,8,18)(19,21,23,28)(20,25,26,34)(22,31,29,27)(24,32,30,33)80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 100343);