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×#31481 |
| 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 | |
1634 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)(13,15)(14,16)(17,18)(20,24)(25,33)(26,31)(27,34);(4,7)(5,10)(11,13,14,17)(12,15,16,18)(19,20)(21,27)(22,24)(23,31)(25,29)(26,30)(28,34)(32,33);(12,16)(15,18)(19,21,22,28)(20,25,24,33)(23,32,30,29)(26,34,31,27);(3,5,6,10)(4,8,7,9)(12,16)(15,18)(20,24)(25,33)(26,31)(27,34)80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7412012);