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×#31655 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×C4×C4⋊C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1954 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);(3,4,8,9)(5,7,10,6)(11,12)(13,15)(14,16)(17,18)(21,28)(25,33)(29,34)(30,32);(3,5,8,10)(4,6,9,7)(11,13,14,17)(12,15,16,18)(19,20)(21,29)(22,26)(23,27)(24,31)(25,30)(28,34)(32,33);(3,6,8,7)(4,5,9,10)(12,16)(15,18)(19,21)(20,25)(22,30)(23,28)(24,32)(26,34)(27,33)(29,31);(3,7,8,6)(4,5,9,10)(12,16)(15,18)(19,22,23,24)(20,26,27,31)(21,30,28,32)(25,34,33,29)80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7412535);