Finite Group DatabasebetaFinite Group Databasebeta| order | 32512 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C127 · Schur multiplier (H2) = C24 |
| direct product | C127×#13035 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C127 | #1879 |
| 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 | |
456 conjugacy classes of subgroups.
| minimal degree | μ(G) = 191 |
| transitive degree | μt(G) = 8128 |
A minimal faithful permutation representation — 3 generators on 191 points:
(1,2,6,13,31,34,30,11,8,15,29,45,7,14,5,12)(3,16,20,9,32,36,18,46,22,49,56,38,21,43,19,37)(4,23,25,42,33,44,24,41,27,39,28,10,26,40,58,47)(17,35,52,60,57,63,51,59,54,62,55,48,53,61,50,64)(65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191);(1,3,7,32,8,22,31,21)(2,9,34,37,15,38,14,46)(4,17,33,53,27,54,26,57)(5,20,29,18,30,56,6,19)(10,35,41,63,42,62,47,61)(11,43,45,49,12,36,13,16)(23,48,40,64,39,60,44,59)(24,55,28,51,58,52,25,50);(1,4)(2,10)(3,17)(5,28)(6,24)(7,33)(8,27)(9,35)(11,40)(12,44)(13,23)(14,47)(15,42)(16,48)(18,52)(19,55)(20,51)(21,57)(22,54)(25,30)(26,31)(29,58)(32,53)(34,41)(36,59)(37,63)(38,62)(39,45)(43,64)(46,61)(49,60)(50,56)5080 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(32512, 6023);