Finite Group DatabasebetaFinite Group Databasebeta| order | 32512 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C127 · Schur multiplier (H2) = C24 |
| direct product | C127×#13322 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C127 | #447 |
| commutator, Frattini, cosocle | C22×C8 | C23×C127 |
| prime radical | C2×#41×C127 | D4 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C127 ◁ C22×C4×C127 ◁ C2×#131×C127 ◁ G | |
520 conjugacy classes of subgroups.
| minimal degree | μ(G) = 195 |
| transitive degree | μt(G) = 16256 |
A minimal faithful permutation representation — 3 generators on 195 points:
(5,6,12,18)(7,19,24,50)(8,25,28,47)(9,29,34,45)(10,17,11,16)(13,36,40,21)(14,35,42,27)(15,32,46,33)(20,51,54,68)(22,43,23,49)(26,44,31,41)(30,38,56,39)(37,60,67,63)(48,57,65,53)(52,58,55,66)(59,64,62,61)(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)(5,7,12,24)(6,13,18,40)(8,20,28,54)(9,30,34,56)(10,23,11,22)(14,37,42,67)(15,43,46,49)(16,39,17,38)(19,45,50,29)(21,32,36,33)(25,58,47,66)(26,60,31,63)(27,57,35,53)(41,68,44,51)(48,59,65,62)(52,64,55,61);(2,4)(5,8)(6,14)(7,20)(9,31)(10,35)(11,27)(12,28)(13,37)(15,44)(16,47)(17,25)(18,42)(19,48)(21,55)(22,57)(23,53)(24,54)(26,34)(29,62)(30,63)(32,61)(33,64)(36,52)(38,58)(39,66)(40,67)(41,46)(43,51)(45,59)(49,68)(50,65)(56,60)5080 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(32512, 6310);