Finite Group DatabasebetaFinite Group Databasebeta| order | 32512 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C127 · Schur multiplier (H2) = C22×C4 |
| direct product | C127×#13216 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C127 | #1961 |
| commutator | C2×C8 | C22×C4×C127 |
| Frattini, cosocle | C22×C8 | C23×C127 |
| prime radical | C2×#135×C127 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C127 ◁ C22×C4×C127 ◁ #405×C127 ◁ G | |
434 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)(4,20,24,53)(5,26)(6,12)(7,30)(8,13)(9,17)(10,23,38,25)(11,27)(15,39)(16,46,48,62)(18,41)(19,43)(21,54,57,40)(22,58,59,37)(28,31)(29,42)(32,56,51,63)(33,49)(34,44)(35,50)(36,47,61,64)(45,60,55,52)(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);(1,3,15,44,7,18,5,17,8,19,27,49,6,29,28,50)(2,9,31,41,12,34,11,14,13,35,26,42,30,33,39,43)(4,21,55,64,25,48,59,63,24,57,45,47,23,16,22,56)(10,36,52,54,20,51,58,46,38,61,60,40,53,32,37,62);(1,4)(2,10,13,38)(3,16,19,48)(5,22)(6,23)(7,25)(8,24)(9,32)(11,37,39,58)(12,20,30,53)(14,40)(15,45)(17,47,50,64)(18,21,29,57)(26,52,31,60)(27,55)(28,59)(33,61)(34,36)(35,51)(41,46)(42,62)(43,54)(44,63,49,56)4318 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(32512, 6204);