Finite Group DatabasebetaFinite Group Databasebeta| order | 32512 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C127 · Schur multiplier (H2) = C23 |
| direct product | C127×#11869 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C127 | #1519 |
| commutator | C2×C8 | C22×C4×C127 |
| Frattini, cosocle | C2×M16 | C23×C127 |
| prime radical | C2×#41×C127 | D4 |
| LCS | G ▷ C2×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C127 ◁ C2×C4×C127 ◁ C2×#131×C127 ◁ G | |
364 conjugacy classes of subgroups; only the 110 normal subgroups are shown.
This group has 364 conjugacy classes of subgroups — too many to draw. Only its 110 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)(3,14,17,45)(4,20)(5,26,29,43)(6,12)(7,32)(8,13)(9,16)(10,24)(11,30,41,31)(15,46,48,62)(18,39,19,53)(21,56,49,44)(22,40,28,37)(23,38)(25,55)(27,42)(33,57)(34,51)(35,52)(36,50)(47,63)(54,58,60,59)(61,64)(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);(1,3,8,17)(2,9,13,35)(4,21,25,49)(5,27,29,51)(6,19,7,18)(10,36,38,63)(11,39,41,53)(12,42,32,34)(14,43,45,26)(15,24,48,23)(16,30,52,31)(20,33,55,61)(22,47,28,50)(37,44,40,56)(46,54,62,60)(57,58,64,59);(1,4,6,23,8,25,7,24)(2,10,32,55,13,38,12,20)(3,15,18,21,17,48,19,49)(5,28,30,59,29,22,31,58)(9,33,42,63,35,61,34,36)(11,40,43,54,41,37,26,60)(14,44,53,62,45,56,39,46)(16,50,51,64,52,47,27,57)4318 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(32512, 4857);