Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30412 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C4 |
| direct product | C2×#32309 |
| 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 | |
1836 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);(3,4,7,10)(5,6,9,8)(11,12,14,16)(13,17,18,15)(19,20)(21,26)(22,27)(23,25)(24,28)(29,34)(30,32)(31,33);(3,5,7,9)(4,8,10,6)(11,13)(12,15)(14,18)(16,17)(20,25)(21,29)(22,24)(26,33)(30,31)(32,34);(3,6,7,8)(4,9,10,5)(12,16)(15,17)(19,21)(20,26)(22,29)(23,30)(24,31)(25,32)(27,34)(28,33);(4,10)(5,9)(13,18)(15,17)(19,22,23,24)(20,27,25,28)(21,29,30,31)(26,34,32,33)80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7415962);