Finite Group DatabasebetaFinite Group Databasebeta| order | 32512 |
| nilpotency class | 5 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C127 · Schur multiplier (H2) = C2 |
| direct product | C127×#13716 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C127 | C2×D16 |
| commutator, Frattini, cosocle | C2×C16 | C23×C127 |
| prime radical | C23×C127 | D16 |
| LCS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C127 ◁ C22×C4×C127 ◁ C22×C8×C127 ◁ C22×C16×C127 ◁ G | |
168 conjugacy classes of subgroups; only the 70 normal subgroups are shown.
This group has 168 conjugacy classes of subgroups — too many to draw. Only its 70 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 199 |
| transitive degree | μt(G) = 16256 |
A minimal faithful permutation representation — 3 generators on 199 points:
(1,2)(3,6)(4,5)(7,8)(9,10,15,21)(11,16,25,38)(12,17,29,41)(13,30,32,59)(14,33,31,20)(18,42,44,26)(19,45,46,27)(22,34,51,62)(23,52,54,72)(24,55,53,37)(28,47,56,40)(35,63,65,48)(36,66,67,49)(39,58,43,57)(50,68,69,61)(60,71,64,70)(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);(2,5)(6,8)(9,11,15,25)(10,16,21,38)(12,22,29,51)(13,23,32,54)(14,24,31,53)(17,34,41,62)(18,35,44,65)(19,36,46,67)(20,37,33,55)(26,48,42,63)(27,49,45,66)(28,50,56,69)(30,52,59,72)(39,60,43,64)(40,61,47,68)(57,70,58,71);(1,3,4,7)(2,6,5,8)(10,17)(12,26)(13,27)(14,31)(16,34)(18,19)(20,47)(21,41)(22,48)(23,49)(24,53)(28,58)(29,42)(30,39)(32,45)(33,40)(35,36)(37,68)(38,62)(43,59)(44,46)(50,71)(51,63)(52,60)(54,66)(55,61)(56,57)(64,72)(65,67)(69,70)6223 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(32512, 6704);