Finite Group DatabasebetaFinite Group Databasebeta| order | 32512 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C127 · Schur multiplier (H2) = C24 |
| direct product | C127×#13069 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C127 | #1893 |
| commutator | C2×C8 | C22×C4×C127 |
| Frattini, cosocle | C22×C8 | C23×C127 |
| prime radical | C2×#41×C127 | D4 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C127 ◁ C22×C4×C127 ◁ C2×#131×C127 ◁ G | |
450 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)(4,20,23,38)(5,24)(6,12)(7,30)(8,13)(9,17)(10,22,37,29)(11,26)(15,40)(16,47,50,63)(18,42)(19,44)(21,36,25,39)(27,31)(28,43)(32,59,62,64)(33,53)(34,45)(35,54)(41,49,60,51)(46,56,57,58)(48,55,52,61)(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);(1,3,15,45,7,18,5,17,8,19,26,53,6,28,27,54)(2,9,31,42,12,34,11,14,13,35,24,43,30,33,40,44)(4,16,46,59,22,51,25,52,23,50,57,64,29,49,21,48)(10,32,58,63,38,61,39,41,37,62,56,47,20,55,36,60);(1,4)(2,10)(3,16)(5,25)(6,29)(7,22)(8,23)(9,32)(11,39)(12,38)(13,37)(14,41)(15,46)(17,52)(18,51)(19,50)(20,30)(21,27)(24,56)(26,57)(28,49)(31,58)(33,55)(34,61)(35,62)(36,40)(42,63)(43,47)(44,60)(45,59)(48,54)(53,64)5842 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(32512, 6057);