Finite Group DatabasebetaFinite Group Databasebeta| order | 32512 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C127 · Schur multiplier (H2) = C25 |
| direct product | C127×#12773 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C127 | B(4,𝔽2) |
| commutator | C2×Q8 | C22×C4×C127 |
| Frattini, cosocle | C2×#41 | C23×C127 |
| socle | C2×C127 | #1790 |
| prime radical | C127×#3483 | C2 |
| LCS | G ▷ C2×Q8 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C127 ◁ C2×C4×C127 ◁ C2×#41×C127 ◁ G | |
832 conjugacy classes of subgroups; only the 138 normal subgroups are shown.
This group has 832 conjugacy classes of subgroups — too many to draw. Only its 138 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 143 |
| transitive degree | μt(G) = 2032 |
A minimal faithful permutation representation — 3 generators on 143 points:
(1,2)(3,7,4,9)(5,10)(6,11)(8,12,16,13)(14,15)(17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143);(2,7)(3,4)(8,10)(9,11)(12,13)(15,16);(1,3)(2,8)(4,6)(5,13)(7,15)(9,10)(11,16)(12,14)5080 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(32512, 5761);