Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #18712 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
| direct product | C2×#18727 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#120 |
| commutator, Frattini, cosocle | C42 | C25 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
3236 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)(5,10)(6,12)(7,11)(8,13)(9,14)(15,18)(16,17)(19,20)(21,29)(22,27)(23,26)(24,28)(25,31)(30,34)(32,33);(3,5)(4,9)(6,14)(7,16)(8,15)(10,11)(12,18)(13,17)(19,21,24,32)(20,25,28,34)(22,30,23,31)(26,33,27,29);(3,5)(4,10)(6,16)(7,14)(8,15)(9,11)(12,17)(13,18)(20,26)(22,23)(25,33)(27,28)(29,34)(30,31);(3,6,8,7)(4,11,13,12)(5,14,15,16)(9,10,17,18)(19,22,24,23)(20,26,28,27)(21,30,32,31)(25,33,34,29)80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6255532);