Finite Group DatabasebetaFinite Group Databasebeta| order | 32512 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C127 · Schur multiplier (H2) = C25 |
| direct product | C127×#12714 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C127 | #1781 |
| commutator | C22×C4 | C22×C4×C127 |
| Frattini, cosocle | C2×#41 | C23×C127 |
| prime radical | C127×#3487 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C127 ◁ C2×C4×C127 ◁ C2×2−1+4×C127 ◁ G | |
974 conjugacy classes of subgroups.
| minimal degree | μ(G) = 159 |
| transitive degree | μt(G) = 4064 |
A minimal faithful permutation representation — 3 generators on 159 points:
(1,2)(3,13)(4,16,18,27)(5,19,21,10)(6,11)(7,12)(8,15,24,20)(9,14,26,22)(17,30)(23,31)(25,28)(29,32)(33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159);(1,3,7,6)(2,8,12,24)(4,14,18,22)(5,15,21,20)(9,23,26,30)(10,13,19,11)(16,29,27,25)(17,28,31,32);(1,4,7,18)(2,9,12,26)(3,14,6,22)(5,17,21,31)(8,23,24,30)(10,25,19,29)(11,27,13,16)(15,28,20,32)4318 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(32512, 5702);