Finite Group DatabasebetaFinite Group Databasebeta| order | 32512 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C127 · Schur multiplier (H2) = C22 |
| direct product | C127×#13529 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23×C127 | C2×D8 |
| commutator, Frattini, cosocle | C22×C8 | C23×C127 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C127 ◁ C2×C42×C127 ◁ C2×C4×C8×C127 ◁ G | |
342 conjugacy classes of subgroups.
| minimal degree | μ(G) = 171 |
| transitive degree | μt(G) = 32512 |
A minimal faithful permutation representation — 3 generators on 171 points:
(5,6,8,10)(7,11,12,9)(13,14,16,21,30,38,41,34,18,23,31,33,17,22,28,19)(15,24,25,39,32,44,43,36,27,40,42,35,26,37,29,20)(45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171);(1,2)(3,4)(5,7,8,12)(6,9,10,11)(14,19)(16,28)(17,30)(20,24)(21,22)(23,34)(25,29)(26,32)(31,41)(33,38)(35,44)(36,40)(37,39)(42,43);(2,4)(13,15,18,27)(14,20,23,36)(16,29,31,43)(17,32,30,26)(19,24,34,40)(21,37,33,44)(22,39,38,35)(25,41,42,28)6604 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(32512, 6517);