Finite Group DatabasebetaFinite Group Databasebeta| order | 32128 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C251 · Schur multiplier (H2) = C2 |
| direct product | #1353×C251 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C251 | #99 |
| commutator | C22×C4 | C2×C4×C251 |
| Frattini, cosocle | C2×C4⋊C4 | C22×C251 |
| prime radical | C23×C251 | C22⋊C4 |
| LCS | G ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C251 ◁ C23×C251 ◁ C2×C4⋊C4×C251 ◁ G | |
116 conjugacy classes of subgroups; only the 40 normal subgroups are shown.
This group has 116 conjugacy classes of subgroups — too many to draw. Only its 40 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 275 |
| transitive degree | μt(G) = 8032 |
A minimal faithful permutation representation — 2 generators on 275 points:
(2,5,6,7)(9,10,12,16,13,15,19,22)(11,14,18,20,17,21,24,23)(25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275);(1,2)(3,7)(4,6)(5,8)(9,11)(10,14,15,21)(12,18)(13,17)(16,23,22,20)(19,24)5773 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(32128, 85);