Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32947 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29×C4 |
| direct product | C2×#32948 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2847 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
4574 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);(4,9)(7,10)(11,19)(12,20)(13,21)(14,27)(15,22)(17,23)(24,33)(25,30)(26,31)(29,34);(3,4,10,21,8,13,7,9)(5,11,22,30,16,25,15,19)(6,12,23,31,18,26,17,20)(14,24,32,33,27,34,28,29);(3,5)(4,11)(6,14)(7,15)(8,16)(9,19)(10,22)(12,24)(13,25)(17,28)(18,27)(20,29)(21,30)(23,32)(26,34)(31,33);(3,6)(4,12)(5,14)(7,17)(8,18)(9,20)(10,23)(11,24)(13,26)(15,28)(16,27)(19,29)(21,31)(22,32)(25,34)(30,33)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6536238);