Finite Group DatabasebetaFinite Group Databasebeta| order | 32512 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C127 · Schur multiplier (H2) = C23 |
| direct product | C127×#12840 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C127 | #427 |
| commutator | C2×C8 | C22×C4×C127 |
| Frattini, cosocle | C22×C8 | C23×C127 |
| socle | C2×C127 | #1808 |
| prime radical | C2×#41×C127 | D4 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C127 ◁ C2×M16×C127 ◁ C2×#131×C127 ◁ G | |
354 conjugacy classes of subgroups; only the 154 normal subgroups are shown.
This group has 354 conjugacy classes of subgroups — too many to draw. Only its 154 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:
(1,2,6,13,8,12,7,29)(3,14,18,41,20,40,19,50)(4,21,23,36,24,10,28,35)(5,25,26,37,27,56,15,11)(9,30,33,48,32,42,39,17)(16,44,46,62,47,38,49,61)(22,53,43,52,54,55,57,34)(31,58,59,45,51,63,60,64)(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);(1,3,15,42,7,19,27,48,8,20,26,30,6,18,5,17)(2,9,25,41,13,33,37,40,12,32,56,50,29,39,11,14)(4,16,43,63,23,46,54,64,24,47,57,58,28,49,22,45)(10,31,55,62,36,60,52,44,21,51,53,61,35,59,34,38);(1,4)(2,10)(3,16)(5,22)(6,28)(7,23)(8,24)(9,31)(11,34)(12,21)(13,36)(14,38)(15,43)(17,45)(18,49)(19,46)(20,47)(25,55)(26,57)(27,54)(29,35)(30,58)(32,51)(33,60)(37,52)(39,59)(40,44)(41,62)(42,63)(48,64)(50,61)(53,56)6604 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(32512, 5828);