Finite Group DatabasebetaFinite Group Databasebeta| order | 32512 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C16×C127 · Schur multiplier (H2) = C23 |
| direct product | C2×C127×#1401 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C16×C127 | D4 |
| commutator | C4 | C22×C16×C127 |
| Frattini, cosocle | C2×C16 | C23×C127 |
| socle | C22×C127 | C22⋊C16 |
| prime radical | C2×#41×C127 | C8 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C16×C127 ◁ C22×C16×C127 ◁ G | |
294 conjugacy classes of subgroups.
| minimal degree | μ(G) = 193 |
| transitive degree | μt(G) = 16256 |
A minimal faithful permutation representation — 3 generators on 193 points:
(1,2)(67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193);(3,4,7,13,8,14,26,39,9,15,27,40,29,42,54,62,10,16,28,41,30,43,55,63,6,12,23,36,24,37,51,60)(5,17,19,44,20,45,48,64,18,34,46,38,47,58,65,61,21,35,25,52,49,59,53,66,22,11,31,32,50,33,56,57);(3,5)(4,11)(6,22)(7,25)(8,20)(9,18)(10,21)(12,35)(13,38)(14,33)(15,17)(16,34)(19,28)(23,46)(24,50)(26,53)(27,31)(29,47)(30,49)(32,41)(36,44)(37,59)(39,61)(40,52)(42,45)(43,58)(48,55)(51,65)(54,56)(57,63)(60,64)(62,66)14224 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(32512, 6545);