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×#12664 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C16×C127 | D4 |
| commutator | C4 | C2×C4×C8×C127 |
| Frattini, cosocle | C22×C8 | C23×C127 |
| socle | C22×C127 | C8×D4 |
| prime radical | C23×C127 | #131 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C16×C127 ◁ C22×C16×C127 ◁ G | |
334 conjugacy classes of subgroups.
| minimal degree | μ(G) = 163 |
| transitive degree | μt(G) = 8128 |
A minimal faithful permutation representation — 3 generators on 163 points:
(5,6,10,15,11,16,28,25,12,17,29,24,9,14,23,8)(7,18,21,19,20,32,35,34,22,33,27,36,26,13,30,31)(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);(1,2,3,4)(5,7)(6,13)(8,19)(9,26)(10,27)(11,20)(12,22)(14,33)(15,34)(16,18)(17,32)(21,29)(23,35)(24,31)(25,36)(28,30);(5,8,23,14,9,24,29,17,12,25,28,16,11,15,10,6)(7,19,35,33,26,31,21,32,22,36,30,18,20,34,27,13)14224 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(32512, 5652);