Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #17603 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C29 |
| direct product | C2×#17776 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×C22⋊C4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C42 | C25 |
| prime radical | C2×#3508 | C2 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
3232 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);(3,4,7,12,8,11,6,10)(5,13,14,18,15,9,16,17)(19,20)(21,30)(22,27)(23,28)(24,33)(25,29)(26,31)(32,34);(3,5,8,15)(4,9,11,13)(6,14,7,16)(10,17,12,18)(19,21,25,22)(20,26,29,34)(23,31,24,32)(27,33,30,28);(3,6,8,7)(4,10,11,12)(5,14,15,16)(9,17,13,18)(19,22,25,21)(20,27,29,30)(23,31,24,32)(26,33,34,28);(3,6,8,7)(4,10,11,12)(5,14,15,16)(9,17,13,18)(19,23,25,24)(20,28,29,33)(21,31,22,32)(26,27,34,30)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6251338);