Finite Group DatabasebetaFinite Group Databasebeta| order | 32512 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C127 · Schur multiplier (H2) = C24 |
| direct product | C127×#8514 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C127 | #379 |
| commutator | C42 | C22×C4×C127 |
| Frattini, cosocle | C2×C42 | C23×C127 |
| prime radical | C127×#3568 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C127 ◁ C2×C42×C127 ◁ G | |
648 conjugacy classes of subgroups; only the 138 normal subgroups are shown.
This group has 648 conjugacy classes of subgroups — too many to draw. Only its 138 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:
(1,2,5,12,8,3,14,18)(4,20,23,56,25,46,42,55)(6,28,26,38,31,54,43,49)(7,13,27,10,32,19,44,16)(9,33,35,60,15,45,34,22)(11,39,41,51,17,50,53,30)(21,57,24,40,59,64,61,52)(29,36,62,37,63,47,58,48)(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)(2,8)(4,15)(5,10)(6,28)(7,19)(9,25)(11,30)(12,27)(13,32)(14,16)(17,51)(18,44)(20,29)(21,34)(22,40)(23,37)(24,47)(26,53)(31,54)(33,55)(35,59)(36,61)(38,50)(39,49)(41,43)(42,48)(45,56)(46,63)(52,60)(57,62)(58,64);(1,4)(2,9)(3,15)(5,23)(6,29)(7,24)(8,25)(10,37)(11,40)(12,35)(13,36)(14,42)(16,48)(17,52)(18,34)(19,47)(20,28)(21,44)(22,30)(26,62)(27,59)(31,63)(32,61)(33,39)(38,56)(41,64)(43,58)(45,50)(46,54)(49,55)(51,60)(53,57)5080 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(32512, 1502);