Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #17597 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C29 |
| direct product | C2×#17627 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×C22⋊C4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| prime radical | C23×D4 | D4 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C4×D4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
3148 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)(5,11,13,8)(6,10)(7,9)(12,17,18,15)(14,16)(19,20)(21,27)(22,30)(23,25)(24,28)(26,32)(29,34)(31,33);(3,5,7,13)(4,8,9,11)(6,12,14,18)(10,15,16,17)(19,21,23,29)(20,24,25,33)(22,31,32,28)(26,34,30,27);(4,9)(6,14)(8,11)(12,18)(20,25)(24,33)(26,30)(27,34);(3,6,7,14)(4,10,9,16)(5,12,13,18)(8,15,11,17)98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6250666);