Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32894 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C212 |
| direct product | C2×#32895 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2846 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
16750 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);(4,10)(5,14)(7,8)(12,27)(13,19)(15,29)(16,21)(17,20)(22,25)(23,28)(24,33)(30,31);(3,4)(5,11)(6,12)(7,19)(8,10)(9,13)(14,25)(15,24)(16,26)(17,23)(18,28)(20,27)(21,22)(29,34)(30,32)(31,33);(3,5,9,16)(4,11,13,26)(6,15,18,31)(7,14,8,21)(10,22,19,25)(12,24,28,33)(17,30,20,29)(23,32,27,34);(3,6)(4,12)(5,15)(7,20)(8,17)(9,18)(10,23)(11,24)(13,28)(14,29)(16,31)(19,27)(21,30)(22,32)(25,34)(26,33)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6536076);