Finite Group DatabasebetaFinite Group Databasebeta| order | 32512 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C127 · Schur multiplier (H2) = C23 |
| direct product | C127×#12827 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C127 | #1804 |
| commutator | C2×C8 | C22×C4×C127 |
| Frattini, cosocle | C22×C8 | C23×C127 |
| prime radical | C23×C127 | AGL(1,ℤ/8) |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C127 ◁ C22×C4×C127 ◁ #405×C127 ◁ G | |
376 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)(3,14,18,33)(4,19)(5,10)(6,26)(7,13)(8,16,32,49)(9,21)(11,24)(12,25)(15,43,46,64)(17,31,27,38)(20,52)(22,53)(23,39)(28,34)(29,37)(30,59,62,50)(35,57)(36,56)(40,55)(41,54,60,45)(42,58,63,48)(44,61,47,51)(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);(1,3,7,18)(2,8)(4,15,22,46)(5,16,25,49)(6,27,29,17)(9,30)(10,14)(11,38)(12,33)(13,32)(19,51)(20,44,23,47)(21,54,56,45)(24,58,55,48)(26,42)(28,59,57,50)(31,40)(34,60)(35,41)(36,62)(37,63)(39,64)(43,52)(53,61);(1,4,7,22)(2,9,13,36)(3,15,18,46)(5,23,25,20)(6,28,29,57)(8,30,32,62)(10,35,12,34)(11,39,40,52)(14,41,33,60)(16,47,49,44)(17,50,27,59)(19,37,53,26)(21,55,56,24)(31,43,38,64)(42,51,63,61)(45,58,54,48)5080 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(32512, 5815);