Finite Group DatabasebetaFinite Group Databasebeta| order | 32512 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C127 · Schur multiplier (H2) = C8 |
| direct product | C127×#12514 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C22×C127 | #493 |
| commutator, Frattini, cosocle | C4×C8 | C23×C127 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C127 ◁ C42×C127 ◁ C82×C127 ◁ G | |
246 conjugacy classes of subgroups.
| minimal degree | μ(G) = 175 |
| transitive degree | μt(G) = 32512 |
A minimal faithful permutation representation — 3 generators on 175 points:
(1,2)(3,9)(4,12)(5,7)(6,10)(8,13)(11,15)(14,16)(17,18,25,32)(19,33,38,27)(20,39,34,26)(21,29,40,48)(22,28,42,46)(23,31,24,30)(35,45,41,44)(36,47,37,43)(49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175);(1,3,4,10,5,11,13,16)(2,6,8,9,7,14,12,15)(17,19,22,35,23,36,40,20,25,38,42,41,24,37,21,34)(18,26,29,43,30,44,46,27,32,39,48,47,31,45,28,33);(2,7)(6,14)(8,12)(9,15)(17,20,40,37,24,35,22,38,25,34,21,36,23,41,42,19)(18,27,46,45,31,43,29,39,32,33,28,44,30,47,48,26)5842 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(32512, 5502);