Finite Group DatabasebetaFinite Group Databasebeta| order | 32128 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C251 · Schur multiplier (H2) = C22 |
| direct product | #2222×C251 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C251 | #466 |
| commutator, Frattini, cosocle | C2×C8 | C23×C251 |
| prime radical | #41×C251 | D4 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C251 ◁ C2×C4×C251 ◁ #131×C251 ◁ G | |
140 conjugacy classes of subgroups; only the 60 normal subgroups are shown.
This group has 140 conjugacy classes of subgroups — too many to draw. Only its 60 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 315 |
| transitive degree | μt(G) = 16064 |
A minimal faithful permutation representation — 3 generators on 315 points:
(1,2,8,14)(3,15,21,43)(4,22,25,39)(5,26,28,37)(6,13,7,12)(9,30,34,18)(10,29,36,24)(11,27,38,16)(17,46,49,63)(19,42,20,41)(23,54,55,35)(31,58,59,64)(32,50,33,44)(40,51,62,48)(45,53,56,61)(47,52,57,60)(65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304,305,306,307,308,309,310,311,312,313,314,315);(1,3,16,44,7,20,28,30,8,21,27,50,6,19,5,18)(2,9,26,42,13,33,38,43,14,34,37,41,12,32,11,15)(4,17,45,64,24,48,55,57,25,49,56,58,29,51,23,47)(10,31,53,46,22,52,54,62,36,59,61,63,39,60,35,40);(1,4)(2,10)(3,17)(5,23)(6,29)(7,24)(8,25)(9,31)(11,35)(12,39)(13,22)(14,36)(15,40)(16,45)(18,47)(19,51)(20,48)(21,49)(26,53)(27,56)(28,55)(30,57)(32,60)(33,52)(34,59)(37,61)(38,54)(41,63)(42,46)(43,62)(44,64)(50,58)6526 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(32128, 954);