Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32696 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
| direct product | C2×#32697 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2749 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1447 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);(3,4,9,13,8,14,26,34)(5,10,18,30,17,29,27,32)(6,19,22,23,21,11,31,16)(7,12,25,15,24,20,28,33);(3,5)(4,10)(6,20)(7,23)(8,17)(9,27)(11,28)(12,21)(13,32)(14,29)(15,22)(16,24)(18,26)(19,25)(30,34)(31,33);(3,6,8,21)(4,11)(5,15)(7,10,24,29)(9,22,26,31)(12,27)(13,16)(14,19)(17,33)(18,20)(23,34)(25,30,28,32);(3,7,8,24)(4,12,14,20)(5,16,17,23)(6,10,21,29)(9,28,26,25)(11,27,19,18)(13,33,34,15)(22,32,31,30)68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7421134);