Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24264 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
| direct product | C2×#29076 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×C4×C4⋊C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1918 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);(3,4,6,7)(5,9,10,8)(11,12,14,16)(13,17,18,15)(19,20,22,25)(21,27,30,24)(23,26,28,32)(29,34,31,33);(4,7)(8,9)(11,13,14,18)(12,15,16,17)(19,21,28,29,22,30,23,31)(20,24,32,33,25,27,26,34);(12,16)(15,17)(21,29)(23,28)(24,33)(26,32)(27,34)(30,31);(3,5,6,10)(4,8,7,9)(12,16)(15,17)(20,25)(24,27)(26,32)(33,34)116 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6367770);