Finite Group DatabasebetaFinite Group Databasebeta| order | 32512 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C127 · Schur multiplier (H2) = C24 |
| direct product | C127×#21372 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C127 | C22×D4 |
| commutator | C2×C4 | C23×C4×C127 |
| Frattini, cosocle | C22×C4 | C24×C127 |
| socle | C22×C127 | C22×C4⋊C4 |
| prime radical | C24×C127 | C2×Q8 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C127 ◁ C2×#117×C127 ◁ G | |
714 conjugacy classes of subgroups.
| minimal degree | μ(G) = 167 |
| transitive degree | μt(G) = 8128 |
A minimal faithful permutation representation — 4 generators on 167 points:
(1,2)(3,6)(4,5)(7,8)(9,10,14,18,15,19,13,27)(11,20,22,16,23,30,28,31)(12,17,26,33,24,32,25,37)(21,34,36,29,35,40,38,39)(41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167);(9,11,14,22,15,23,13,28)(10,16,18,30,19,31,27,20)(12,21,26,36,24,35,25,38)(17,29,33,40,32,39,37,34);(2,5)(6,8)(9,12)(10,17)(11,21)(13,25)(14,26)(15,24)(16,29)(18,33)(19,32)(20,34)(22,36)(23,35)(27,37)(28,38)(30,40)(31,39);(1,3,4,7)(2,6,5,8)(12,24)(17,32)(21,35)(25,26)(29,39)(33,37)(34,40)(36,38)8128 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(32512, 14360);