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×#12839 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C127 | #427 |
| 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 ◁ C2×M16×C127 ◁ C2×#131×C127 ◁ G | |
362 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)(3,13)(4,19,21,9)(5,23)(6,26)(7,11)(8,16)(10,14)(12,24)(15,40,42,33)(17,46)(18,35)(20,50,52,53)(22,31,51,49)(25,54)(27,41,47,63)(28,45)(29,56)(30,38,48,60)(32,39)(34,37)(36,44)(43,58,62,61)(55,64,57,59)(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);(1,3,14,37,6,17,25,45,7,18,39,56,24,44,5,16)(2,8,23,36,12,29,32,35,11,28,54,46,26,34,10,13)(4,15,38,59,22,43,52,63,21,42,60,64,51,62,20,41)(9,27,53,58,31,55,48,40,19,47,50,61,49,57,30,33);(1,4)(2,9)(3,15)(5,20)(6,22)(7,21)(8,27)(10,30)(11,19)(12,31)(13,33)(14,38)(16,41)(17,43)(18,42)(23,53)(24,51)(25,52)(26,49)(28,47)(29,55)(32,48)(34,57)(35,40)(36,58)(37,59)(39,60)(44,62)(45,63)(46,61)(50,54)(56,64)6604 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(32512, 5827);