Finite Group DatabasebetaFinite Group Databasebeta| order | 32512 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C127 · Schur multiplier (H2) = C24 |
| direct product | C127×#13052 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C127 | B(4,𝔽2) |
| commutator | C2×D4 | C22×C4×C127 |
| Frattini, cosocle | C2×#41 | C23×C127 |
| socle | C2×C127 | #1889 |
| prime radical | C127×#3018 | C2 |
| LCS | G ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C127 ◁ C2×C4×C127 ◁ C2×#41×C127 ◁ G | |
624 conjugacy classes of subgroups; only the 114 normal subgroups are shown.
This group has 624 conjugacy classes of subgroups — too many to draw. Only its 114 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 159 |
| transitive degree | μt(G) = 4064 |
A minimal faithful permutation representation — 3 generators on 159 points:
(1,2,6,11,4,9,13,22)(3,12,16,28,14,7,21,24)(5,10,19,27,17,8,20,26)(15,25,29,32,18,23,30,31)(33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159);(1,3,13,21,4,14,6,16)(2,7,22,28,9,12,11,24)(5,18,19,30,17,15,20,29)(8,25,26,32,10,23,27,31);(2,8)(3,14)(5,17)(7,23)(9,10)(11,27)(12,25)(16,21)(19,20)(22,26)(24,32)(28,31)4318 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(32512, 6040);