Finite Group DatabasebetaFinite Group Databasebeta| order | 32512 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C127 · Schur multiplier (H2) = C23 |
| direct product | C127×#12841 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C127 | #466 |
| commutator | C2×C8 | C22×C4×C127 |
| Frattini, cosocle | C22×C8 | C23×C127 |
| socle | C2×C127 | #1808 |
| prime radical | C2×#41×C127 | D4 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C127 ◁ C22×C4×C127 ◁ C2×#131×C127 ◁ G | |
358 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)(3,14)(4,21,24,35)(5,25)(6,12)(7,29)(8,13)(9,17)(10,23,34,28)(11,15)(16,43,46,62)(18,39)(19,50)(20,40)(22,33,53,60)(26,36)(27,55)(30,57,58,64)(31,47)(32,48)(37,45,61,49)(38,41)(42,52,56,54)(44,51,63,59)(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);(1,3,15,41,7,19,27,48,8,20,26,47,6,18,5,17)(2,9,25,39,12,31,36,40,13,32,55,50,29,38,11,14)(4,16,42,57,23,45,53,63,24,46,56,64,28,49,22,44)(10,30,54,62,35,59,60,61,34,58,52,43,21,51,33,37);(1,4)(2,10)(3,16)(5,22)(6,28)(7,23)(8,24)(9,30)(11,33)(12,35)(13,34)(14,37)(15,42)(17,44)(18,49)(19,45)(20,46)(21,29)(25,54)(26,56)(27,53)(31,59)(32,58)(36,60)(38,51)(39,62)(40,61)(41,57)(43,50)(47,64)(48,63)(52,55)6604 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(32512, 5829);