Finite Group DatabasebetaFinite Group Databasebeta| order | 32512 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C127 · Schur multiplier (H2) = C23 |
| direct product | C127×#11709 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C127 | #102 |
| commutator | C2×C8 | C22×C4×C127 |
| Frattini, cosocle | C22×C8 | C23×C127 |
| socle | C22×C127 | C2×#102 |
| prime radical | C127×#3050 | C2 |
| LCS | G ▷ C2×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C127 ◁ C23×C4×C127 ◁ C23×C8×C127 ◁ G | |
450 conjugacy classes of subgroups.
| minimal degree | μ(G) = 167 |
| transitive degree | μt(G) = 8128 |
A minimal faithful permutation representation — 3 generators on 167 points:
(1,2)(3,7,8,5)(4,6)(9,10,12,17,15,20,14,19)(11,21,24,35,25,38,22,37)(13,27,26,18,30,34,29,36)(16,28,32,23,33,39,31,40)(41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167);(1,3)(2,5)(4,8)(6,7)(9,11)(10,16)(12,22)(13,28)(14,24)(15,25)(17,31)(18,35)(19,32)(20,33)(21,34)(23,29)(26,40)(27,38)(30,39)(36,37);(9,12,15,14)(10,17,20,19)(11,22,25,24)(13,26,30,29)(16,31,33,32)(18,34,36,27)(21,37,38,35)(23,28,40,39)7366 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(32512, 4697);