Finite Group DatabasebetaFinite Group Databasebeta| order | 32512 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C127 · Schur multiplier (H2) = C27 |
| direct product | C127×#20935 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C127 | C22×D4 |
| commutator | C2×C4 | C23×C4×C127 |
| Frattini, cosocle | C22×C4 | C24×C127 |
| prime radical | C24×C127 | C2×D4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C127 ◁ C2×#117×C127 ◁ G | |
1690 conjugacy classes of subgroups.
| minimal degree | μ(G) = 167 |
| transitive degree | μt(G) = 16256 |
A minimal faithful permutation representation — 4 generators on 167 points:
(2,4)(5,6)(7,8)(9,10,14,20,15,18,13,19)(11,21,25,16,23,31,24,32)(12,26,28,17,29,34,27,33)(22,30,38,35,36,40,37,39)(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);(1,2,3,4)(9,11)(10,16)(12,22)(13,25)(14,24)(15,23)(17,30)(18,32)(19,31)(20,21)(26,35)(27,38)(28,37)(29,36)(33,40)(34,39);(6,8)(9,12,15,29)(10,17,18,33)(11,22,23,36)(13,28,14,27)(16,30,32,40)(19,34,20,26)(21,35,31,39)(24,38,25,37);(10,18)(11,23)(17,33)(19,20)(22,36)(24,25)(26,34)(37,38)8128 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(32512, 13923);