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×#17625 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×C22⋊C4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#3505 | C2 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C4×D4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
3240 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)(6,14)(7,10)(9,15)(12,16,18,17)(19,20)(21,28)(22,27)(23,31)(24,26)(25,29)(30,34)(32,33);(3,5)(4,8)(6,12)(7,13)(9,16)(10,11)(14,17)(15,18)(19,21)(20,25)(22,29)(23,32)(24,30)(26,33)(27,34)(28,31);(3,6)(4,9)(5,12)(7,15)(8,16)(10,14)(11,17)(13,18)(20,26)(25,33)(27,31)(28,34);(6,15)(9,14)(12,18)(16,17)(19,22,24,23)(20,27,26,31)(21,29,30,32)(25,34,33,28)98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6250664);