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×#17623 |
| 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 | |
3288 conjugacy classes of subgroups.
| minimal degree | μ(G) = 34 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 34 points:
(1,2);(4,10,13,27)(5,16)(7,20)(11,29,30,32)(12,26,25,31)(14,28,15,23)(17,33)(19,21);(3,4)(5,11)(6,12)(7,23)(8,14)(9,15)(10,17)(13,24)(16,26)(18,32)(19,31)(20,27)(21,30)(22,29)(25,34)(28,33);(3,5)(4,11)(6,20)(7,18)(8,16)(9,19)(10,25)(12,27)(13,30)(14,26)(15,31)(17,34)(21,24)(22,33)(23,32)(28,29);(3,6,8,18)(4,12,14,32)(5,17,16,33)(7,21,20,19)(9,22,24,34)(10,26,28,11)(13,25,15,29)(23,30,27,31)98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6250661);