Finite Group DatabasebetaFinite Group Databasebeta| order | 32512 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42×C127 · Schur multiplier (H2) = C2×C4 |
| direct product | C127×#12597 |
| subgroup | G/H | |
|---|---|---|
| center | C8×C127 | #117 |
| commutator | C2×C4 | C2×C42×C127 |
| Frattini, cosocle | C22×C8 | C23×C127 |
| socle | C2×C127 | #1746 |
| prime radical | C23×C127 | #131 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8×C127 ◁ #405×C127 ◁ G | |
258 conjugacy classes of subgroups.
| minimal degree | μ(G) = 191 |
| transitive degree | μt(G) = 8128 |
A minimal faithful permutation representation — 3 generators on 191 points:
(1,2,5,10,6,12,22,34,7,11,23,33,24,35,49,59)(3,13,16,37,17,38,42,61,15,8,25,27,41,28,51,54)(4,9,19,30,21,31,47,57,20,32,46,58,48,52,63,64)(14,36,40,60,18,43,45,62,39,26,50,53,44,29,55,56)(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)(2,8,11,13)(4,18,20,44)(5,16)(6,17)(7,15)(9,29)(10,27,33,37)(12,28,35,38)(14,21,39,48)(19,45,46,55)(22,42)(23,25)(24,41)(26,52)(30,56)(31,36)(32,43)(34,54,59,61)(40,47,50,63)(49,51)(53,64)(57,60)(58,62);(1,4)(2,9)(3,14,15,39)(5,19)(6,21)(7,20)(8,26,13,36)(10,30)(11,32)(12,31)(16,40,25,50)(17,18,41,44)(22,47)(23,46)(24,48)(27,53,37,60)(28,29,38,43)(33,58)(34,57)(35,52)(42,45,51,55)(49,63)(54,56,61,62)(59,64)8128 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(32512, 5585);