Finite Group DatabasebetaFinite Group Databasebeta| order | 32512 |
| nilpotency class | 5 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C127 · Schur multiplier (H2) = C4 |
| direct product | C127×#13721 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C127 | C2×D16 |
| commutator, Frattini, cosocle | C2×C16 | C23×C127 |
| prime radical | C2×D16×C127 | C22 |
| LCS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C127 ◁ C42×C127 ◁ C4×C8×C127 ◁ C4×C16×C127 ◁ G | |
180 conjugacy classes of subgroups; only the 78 normal subgroups are shown.
This group has 180 conjugacy classes of subgroups — too many to draw. Only its 78 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)(11,25)(12,18)(13,35)(14,39)(15,41)(16,44)(17,24)(19,30)(20,26)(21,38)(22,36)(23,42)(27,49)(28,60)(29,50)(31,55)(32,54)(33,66)(34,48)(37,52)(40,51)(43,70)(45,64)(46,63)(47,65)(53,69)(56,61)(57,68)(58,67)(59,72)(62,71)(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)(9,11,26,32,15,28,57,65,42,61,40,59,69,64,38,34,17,30,14,27,43,62,37,33,16,29,58,63,36,31,13,12)(10,18,35,55,22,46,67,50,44,66,52,71,70,49,39,19,24,48,21,45,53,72,51,56,23,47,68,60,41,54,20,25);(1,3,4,7)(2,6,5,8)(9,12,13,31,36,63,58,29,16,33,37,62,43,27,14,30,17,34,38,64,69,59,40,61,42,65,57,28,15,32,26,11)(10,19,20,49,41,71,68,66,23,50,51,46,53,55,21,18,24,25,39,54,70,60,52,47,44,56,67,72,22,45,35,48)8890 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(32512, 6709);