Finite Group DatabasebetaFinite Group Databasebeta| order | 32512 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C127 · Schur multiplier (H2) = C24 |
| direct product | C127×#13335 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C127 | #2011 |
| commutator, Frattini, cosocle | C22×C8 | C23×C127 |
| prime radical | C127×#3414 | C2 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C127 ◁ C22×C4×C127 ◁ #405×C127 ◁ G | |
536 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)(4,18)(5,23)(6,29)(7,11)(8,15)(9,20)(10,27)(12,26)(14,32)(16,54)(17,46)(19,37)(21,59)(22,36)(24,42)(25,43)(28,44)(30,39)(31,40)(33,53)(34,60)(35,50)(38,58)(41,51)(45,48)(47,56)(49,64)(52,62)(55,63)(57,61)(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);(1,3,7,17)(2,8,11,33)(4,14,22,50)(5,24,28,60)(6,30,26,16)(9,35,37,32)(10,39,41,54)(12,42,29,34)(13,44,46,23)(15,51,53,27)(18,56,36,45)(19,57,20,52)(21,55,31,49)(25,48,58,47)(38,63,43,64)(40,61,59,62);(1,4)(2,9)(3,14)(5,25)(6,31)(7,22)(8,32)(10,40)(11,37)(12,43)(13,45)(15,52)(16,55)(17,50)(18,23)(19,51)(20,27)(21,26)(24,48)(28,58)(29,38)(30,49)(33,35)(34,64)(36,44)(39,62)(41,59)(42,63)(46,56)(47,60)(53,57)(54,61)4318 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(32512, 6323);