Finite Group DatabasebetaFinite Group Databasebeta| order | 32512 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C127 · Schur multiplier (H2) = C22 |
| direct product | C127×#12305 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C127 | #1616 |
| commutator, Frattini, cosocle | C2×M16 | C23×C127 |
| prime radical | C23×C127 | C2×SD16 |
| LCS | G ▷ C2×M16 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C127 ◁ C2×C4×C127 ◁ C2×M16×C127 ◁ G | |
262 conjugacy classes of subgroups; only the 70 normal subgroups are shown.
This group has 262 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) = 191 |
| transitive degree | μt(G) = 8128 |
A minimal faithful permutation representation — 3 generators on 191 points:
(1,2,6,26,7,12,28,11)(3,13,17,35,16,40,27,41)(4,18,21,34,20,9,32,48)(5,22,25,10,24,54,55,37)(8,23,38,15,31,45,30,46)(14,39,43,63,44,57,53,60)(19,49,51,36,52,61,59,33)(29,56,62,64,47,50,58,42)(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);(1,3,7,16)(2,8,12,31)(4,14,20,44)(5,23,24,45)(6,27,28,17)(9,29,18,47)(10,35,37,41)(11,38,26,30)(13,22,40,54)(15,25,46,55)(19,50,52,56)(21,53,32,43)(33,60,36,63)(34,62,48,58)(39,49,57,61)(42,51,64,59);(1,4)(2,9,12,18)(3,14)(5,19)(6,21)(7,20)(8,29,31,47)(10,36,37,33)(11,34,26,48)(13,39,40,57)(15,42)(16,44)(17,43)(22,49,54,61)(23,50)(24,52)(25,51)(27,53)(28,32)(30,58,38,62)(35,63,41,60)(45,56)(46,64)(55,59)3175 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(32512, 5293);