Finite Group DatabasebetaFinite Group Databasebeta| order | 32512 |
| nilpotency class | 5 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C127 · Schur multiplier (H2) = C22 |
| direct product | C127×#13655 |
| subgroup | G/H | |
|---|---|---|
| center | C8×C127 | D16 |
| commutator | C16 | C22×C4×C127 |
| Frattini, cosocle | C2×C16 | C23×C127 |
| socle | C2×C127 | C4×D16 |
| prime radical | C127×#2262 | C2 |
| LCS | G ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8×C127 ◁ C2×C8×C127 ◁ C4×C8×C127 ◁ C4×C16×C127 ◁ G | |
186 conjugacy classes of subgroups; only the 86 normal subgroups are shown.
This group has 186 conjugacy classes of subgroups — too many to draw. Only its 86 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:
(1,2,6,12,8,14,29,13)(3,15,18,45,20,46,19,51)(4,10,22,37,23,38,7,30)(5,24,26,54,28,56,60,55)(9,32,34,50,36,49,43,17)(11,21,40,52,42,53,41,27)(16,44,47,63,48,64,31,39)(25,33,57,61,59,62,58,35)(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)(2,9)(4,16)(5,25)(6,18)(7,31)(8,20)(10,33)(11,39)(12,34)(13,43)(14,36)(15,24)(17,27)(19,29)(21,32)(22,47)(23,48)(26,57)(28,59)(30,35)(37,61)(38,62)(40,44)(41,64)(42,63)(45,54)(46,56)(49,53)(50,52)(51,55)(58,60);(1,4,6,22,8,23,29,7)(2,10,12,37,14,38,13,30)(3,16,18,47,20,48,19,31)(5,21,26,52,28,53,60,27)(9,33,34,61,36,62,43,35)(11,24,40,54,42,56,41,55)(15,44,45,63,46,64,51,39)(17,25,32,57,50,59,49,58)9652 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(32512, 6643);