Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #18712 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
| direct product | C2×#18744 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#120 |
| commutator, Frattini, cosocle | C42 | C25 |
| prime radical | C2×#3212 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
2750 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)(5,11)(6,8)(7,15)(9,13)(10,17)(12,16)(14,18)(20,25)(21,28)(22,23)(27,31)(29,30)(32,33);(3,5,8,14)(4,9,6,12)(7,16,17,13)(10,18,15,11)(19,20,24,27)(21,26,30,34)(22,25,23,31)(28,32,29,33);(3,6,8,4)(5,12,14,9)(7,15,17,10)(11,13,18,16)(19,21,24,30)(20,26,27,34)(22,28,23,29)(25,32,31,33);(4,6)(9,12)(10,15)(11,18)(19,22,24,23)(20,25,27,31)(21,29,30,28)(26,33,34,32)80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6255551);