Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30412 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C25×C42 |
| direct product | C2×#32127 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C22×#121 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×C4×C22⋊C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2042 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);(4,7)(5,9)(11,12)(13,17)(14,16)(15,18)(19,20,22,25)(21,27,29,24)(23,26,31,34)(28,33,30,32);(3,4,6,7)(5,10,9,8)(19,21)(20,24)(22,29)(23,28)(25,27)(26,32)(30,31)(33,34);(11,13,14,18)(12,15,16,17)(21,28)(23,31)(24,32)(26,34)(27,33)(29,30);(3,5,6,9)(4,8,7,10)(12,16)(15,17)(21,29)(24,27)(28,30)(32,33)80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7414949);