Finite Group DatabasebetaFinite Group Databasebeta| order | 32512 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C16×C127 · Schur multiplier (H2) = C22 |
| direct product | C127×#13628 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C16×C127 | C23 |
| commutator | C22 | C22×C16×C127 |
| Frattini, cosocle | C2×C16 | C23×C127 |
| socle | C22×C127 | C22×C16 |
| prime radical | C23×C127 | C2×C16 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C16×C127 ◁ G | |
206 conjugacy classes of subgroups.
| minimal degree | μ(G) = 195 |
| transitive degree | μt(G) = 16256 |
A minimal faithful permutation representation — 3 generators on 195 points:
(2,4)(5,6,8,13,9,14,21,31,10,15,22,32,24,34,43,53,11,16,23,33,25,35,44,54,26,36,45,55,46,56,61,66)(7,12,18,28,19,29,40,50,20,30,41,51,42,52,60,65,17,27,37,47,38,48,57,62,39,49,58,63,59,64,67,68)(69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195);(1,2,3,4)(5,7)(6,12)(8,18)(9,19)(10,20)(11,17)(13,28)(14,29)(15,30)(16,27)(21,40)(22,41)(23,37)(24,42)(25,38)(26,39)(31,50)(32,51)(33,47)(34,52)(35,48)(36,49)(43,60)(44,57)(45,58)(46,59)(53,65)(54,62)(55,63)(56,64)(61,67)(66,68);(7,17)(12,27)(18,37)(19,38)(20,39)(28,47)(29,48)(30,49)(40,57)(41,58)(42,59)(50,62)(51,63)(52,64)(60,67)(65,68)14224 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(32512, 6616);