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×#17624 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×C22⋊C4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#532 | C22 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C4×Q8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
2908 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)(5,13)(6,11)(7,10)(8,12)(9,14)(15,17)(16,18)(20,24,26,29)(25,31,32,34);(3,5)(4,9)(6,14)(7,16)(8,15)(10,13)(11,17)(12,18)(19,20)(21,25)(22,29)(23,26)(24,30)(27,32)(28,34)(31,33);(3,6,8,7)(4,10,12,11)(5,14,15,16)(9,13,18,17)(19,21)(20,25)(22,28)(23,27)(24,31)(26,32)(29,34)(30,33);(3,6,8,7)(4,11,12,10)(5,14,15,16)(9,17,18,13)(21,27)(25,32)(28,33)(31,34)98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6250662);