Finite Group DatabasebetaFinite Group Databasebeta| order | 32512 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8×C127 · Schur multiplier (H2) = C22 |
| direct product | C127×#11620 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C127 | C2×C22⋊C4 |
| commutator | C2×C4 | C22×C8×C127 |
| Frattini, cosocle | C4×C8 | C23×C127 |
| socle | C22×C127 | C2×C22⋊C8 |
| prime radical | #121×C127 | C8 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C127 ◁ C2×C4×C8×C127 ◁ G | |
318 conjugacy classes of subgroups.
| minimal degree | μ(G) = 167 |
| transitive degree | μt(G) = 16256 |
A minimal faithful permutation representation — 3 generators on 167 points:
(1,2,5,8,6,3,4,7)(9,10,13,19,14,20,28,37,15,17,29,33,12,18,26,35)(11,21,24,38,23,32,39,36,22,34,27,40,25,16,30,31)(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);(1,3)(2,6)(4,8)(5,7)(9,11)(10,16)(12,25)(13,27)(14,23)(15,22)(17,32)(18,34)(19,36)(20,21)(24,29)(26,39)(28,30)(31,33)(35,38)(37,40);(1,4,6,5)(2,7,3,8)(10,17)(11,22)(18,20)(19,33)(23,25)(24,27)(30,39)(35,37)8128 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(32512, 4608);