Finite Group DatabasebetaFinite Group Databasebeta| order | 32512 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C8×C127 · Schur multiplier (H2) = C22×C4 |
| direct product | C127×#12613 |
| subgroup | G/H | |
|---|---|---|
| center | C16×C127 | C22⋊C4 |
| commutator | C22 | C2×C4×C8×C127 |
| Frattini, cosocle | C22×C8 | C23×C127 |
| socle | C2×C127 | C8×C22⋊C4 |
| prime radical | C2×#41×C127 | C2×C4 |
| LCS | G ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C16×C127 ◁ C22×C16×C127 ◁ G | |
410 conjugacy classes of subgroups.
| minimal degree | μ(G) = 191 |
| transitive degree | μt(G) = 8128 |
A minimal faithful permutation representation — 3 generators on 191 points:
(1,2,5,10,6,12,21,32,7,11,22,31,23,33,45,54)(3,13,16,35,17,36,41,57,15,8,24,26,40,27,47,50)(4,9,18,28,19,30,42,52,20,29,43,51,44,53,60,63)(14,34,38,55,39,56,59,64,37,25,46,48,58,49,61,62)(65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191);(1,3)(2,8,11,13)(4,14)(5,16)(6,17)(7,15)(9,25,29,34)(10,26,31,35)(12,27,33,36)(18,38)(19,39)(20,37)(21,41)(22,24)(23,40)(28,48,51,55)(30,49,53,56)(32,50,54,57)(42,59)(43,46)(44,58)(45,47)(52,62,63,64)(60,61);(1,4,5,18,6,19,21,42,7,20,22,43,23,44,45,60)(2,9,10,28,12,30,32,52,11,29,31,51,33,53,54,63)(3,14,16,38,17,39,41,59,15,37,24,46,40,58,47,61)(8,25,26,48,27,49,50,62,13,34,35,55,36,56,57,64)11176 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(32512, 5601);