Finite Group DatabasebetaFinite Group Databasebeta| order | 32512 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C8×C127 · Schur multiplier (H2) = C2×C4 |
| direct product | C127×#12656 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C8×C127 | C2×D4 |
| commutator | C4 | C2×C4×C8×C127 |
| Frattini, cosocle | C22×C8 | C23×C127 |
| socle | C22×C127 | C2×C4⋊C8 |
| prime radical | C23×C127 | C2×M16 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C8×C127 ◁ C2×C4×C8×C127 ◁ G | |
358 conjugacy classes of subgroups.
| minimal degree | μ(G) = 163 |
| transitive degree | μt(G) = 8128 |
A minimal faithful permutation representation — 3 generators on 163 points:
(1,2,4,3)(5,6)(7,19,22,32)(8,24)(9,15)(10,16)(11,17)(12,18)(13,21,29,27)(14,26)(20,30,34,33)(23,28,35,36)(25,31)(37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163);(5,7,10,23,11,21,26,34,12,22,25,35,9,27,8,20)(6,13,16,30,17,19,14,28,18,29,31,33,15,32,24,36);(1,3,4,2)(5,8,9,25,12,26,11,10)(6,14,15,16,18,24,17,31)(7,20,27,35,22,34,21,23)(13,28,32,30,29,36,19,33)11176 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(32512, 5644);