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×#17761 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×C22⋊C4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C42 | C25 |
| prime radical | C22×#41 | D4 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
2930 conjugacy classes of subgroups.
| minimal degree | μ(G) = 34 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 34 points:
(1,2);(3,4)(5,12)(6,8)(7,11)(9,13)(10,15)(14,17)(16,18)(19,20,23,28,24,26,22,27)(21,29,31,25,32,34,30,33);(4,8)(6,15)(9,16)(10,11)(12,17)(14,18)(19,21,24,32)(20,25,26,33)(22,31,23,30)(27,34,28,29);(3,5)(4,9)(6,14)(7,13)(8,16)(10,17)(11,12)(15,18)(20,26)(25,33)(27,28)(29,34);(5,13)(9,12)(14,18)(16,17)98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6251257);