Finite Group DatabasebetaFinite Group Databasebeta| order | 32512 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C127 · Schur multiplier (H2) = C23 |
| direct product | C2×C127×#1350 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C127 | #102 |
| commutator | C2×C8 | C22×C4×C127 |
| Frattini, cosocle | C22×C8 | C23×C127 |
| prime radical | C24×C127 | SD16 |
| LCS | G ▷ C2×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C127 ◁ C23×C4×C127 ◁ C23×C8×C127 ◁ G | |
446 conjugacy classes of subgroups.
| minimal degree | μ(G) = 169 |
| transitive degree | μt(G) = 16256 |
A minimal faithful permutation representation — 3 generators on 169 points:
(1,2)(43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169);(3,4)(5,9,10,7)(6,8)(11,12,15,20,17,22,16,21)(13,23,25,36,27,40,26,39)(14,28,30,19,32,37,31,38)(18,29,33,24,35,41,34,42);(3,5)(4,7)(6,10)(8,9)(11,13,17,27)(12,18,22,35)(14,29,32,41)(15,26,16,25)(19,36,38,39)(20,34,21,33)(23,28,40,37)(24,30,42,31)7366 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(32512, 4696);