Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2069 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
| direct product | C2×#23622 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #396 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| socle | C22 | C2×#396 |
| prime radical | C23×#41 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3066 conjugacy classes of subgroups.
| minimal degree | μ(G) = 34 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 34 points:
(1,2);(3,4,9,13,8,14,7,12)(5,15,23,26,18,25,17,10)(6,11,20,27,22,28,21,29)(16,30,31,24,19,33,32,34);(3,5,7,17,8,18,9,23)(4,10,13,15,14,26,12,25)(6,19,21,31,22,16,20,32)(11,24,27,33,28,34,29,30);(3,6)(4,11)(5,16)(7,20)(8,22)(9,21)(10,24)(12,27)(13,29)(14,28)(15,30)(17,31)(18,19)(23,32)(25,33)(26,34);(3,7,8,9)(4,12,14,13)(5,17,18,23)(6,20,22,21)(10,25,26,15)(11,27,28,29)(16,31,19,32)(24,33,34,30)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6339347);