Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23860 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
| direct product | C2×#27323 |
| 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 | |
2054 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);(3,4,6,7)(5,8,9,10)(11,12)(13,16)(14,18)(15,17)(19,20,24,28)(21,25,31,34)(22,26,23,27)(29,32,30,33);(4,7)(8,10)(11,13,15,14)(12,16,17,18)(19,21,24,31)(20,25,28,34)(22,30,23,29)(26,33,27,32);(3,5,6,9)(4,8,7,10)(12,17)(16,18)(21,29)(22,23)(25,32)(26,27)(30,31)(33,34);(4,7)(8,10)(11,14,15,13)(12,18,17,16)(19,22,24,23)(20,26,28,27)(21,30,31,29)(25,33,34,32)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6348068);