Finite Group DatabasebetaFinite Group Databasebeta| order | 32896 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C257 · Schur multiplier (H2) = C24 |
| direct product | #2020×C257 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C257 | #121 |
| commutator, Frattini, cosocle | C22×C4 | C23×C257 |
| prime radical | C2×#41×C257 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C257 ◁ C22×C4×C257 ◁ G | |
342 conjugacy classes of subgroups; only the 92 normal subgroups are shown.
This group has 342 conjugacy classes of subgroups — too many to draw. Only its 92 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 293 |
| transitive degree | μt(G) = 16448 |
A minimal faithful permutation representation — 3 generators on 293 points:
(1,2)(3,4)(5,6,11,16)(7,17,21,28)(8,22,26,31)(9,15,10,14)(12,20,29,19)(13,25,32,24)(18,33,35,27)(23,30,34,36)(37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293);(2,4)(5,7)(6,12)(8,23)(9,20)(10,19)(11,21)(13,30)(14,28)(15,17)(16,29)(18,25)(22,33)(24,35)(26,34)(27,31)(32,36);(5,8,11,26)(6,13,16,32)(7,18,21,35)(9,25,10,24)(12,27,29,33)(14,31,15,22)(17,30,28,36)(19,23,20,34)6682 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(32896, 752);