Finite Group DatabasebetaFinite Group Databasebeta| order | 32512 |
| nilpotency class | 5 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C127 · Schur multiplier (H2) = C22 |
| direct product | C127×#13691 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C127 | #2206 |
| commutator, Frattini, cosocle | C2×C16 | C23×C127 |
| prime radical | C127×#2263 | C2 |
| LCS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C127 ◁ C2×C4×C127 ◁ C4×C8×C127 ◁ C4×C16×C127 ◁ G | |
200 conjugacy classes of subgroups; only the 70 normal subgroups are shown.
This group has 200 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)(3,13)(4,19)(5,24)(6,27)(7,31)(8,14)(9,16)(10,21)(11,26)(12,29)(15,44)(17,48)(18,43)(20,53)(22,52)(23,54)(25,34)(28,42)(30,60)(32,40)(33,57)(35,55)(36,61)(37,49)(38,56)(39,50)(41,47)(45,51)(46,64)(58,59)(62,63)(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);(1,3,5,16,8,18,26,7)(2,9,11,13,14,31,24,43)(4,15,20,45,23,46,22,32)(6,17,25,47,30,50,58,29)(10,33,35,42,38,36,37,63)(12,34,39,27,41,59,48,60)(19,51,52,44,54,40,53,64)(21,28,49,57,56,62,55,61);(1,4)(2,10)(3,15)(5,20)(6,28)(7,32)(8,23)(9,33)(11,35)(12,40)(13,42)(14,38)(16,45)(17,49)(18,46)(19,27)(21,29)(22,26)(24,37)(25,57)(30,62)(31,36)(34,53)(39,64)(41,51)(43,63)(44,48)(47,56)(50,55)(52,59)(54,60)(58,61)4699 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(32512, 6679);