Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30412 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
| direct product | C2×#32492 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C24 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×C4×C4⋊C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1538 conjugacy classes of subgroups.
| minimal degree | μ(G) = 34 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 34 points:
(1,2);(3,4,6,7)(5,9,10,8)(11,12,15,17)(13,16,14,18)(19,20)(21,24)(22,26)(23,25)(27,31)(28,33)(29,32)(30,34);(4,7)(5,10)(11,13,15,14)(12,16,17,18)(19,21,23,29)(20,24,25,32)(22,27,30,28)(26,31,34,33);(11,14,15,13)(12,16,17,18)(21,27)(22,30)(24,31)(26,34)(28,29)(32,33);(3,5,6,10)(4,8,7,9)(12,17)(16,18)(20,25)(24,32)(26,34)(31,33)80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7417045);