Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #19456 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
| direct product | C2×#19466 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#121 |
| commutator, Frattini, cosocle | C42 | C25 |
| prime radical | C23×D4 | D4 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
1748 conjugacy classes of subgroups.
| minimal degree | μ(G) = 34 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 34 points:
(1,2);(3,4)(5,12)(6,10)(7,16)(8,11)(9,13)(14,17)(15,18)(19,20,23,27)(21,24,28,32)(22,25,30,26)(29,31,34,33);(3,5,7,13,8,14,6,15)(4,9,10,12,11,18,16,17)(19,21)(20,24)(22,29)(23,28)(25,31)(26,33)(27,32)(30,34);(20,25)(22,30)(24,31)(26,27)(29,34)(32,33);(3,6,8,7)(4,10,11,16)(5,13,14,15)(9,17,18,12)(21,28)(24,32)(29,34)(31,33)80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6261728);