Finite Group DatabasebetaFinite Group Databasebeta| order | 32512 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C127 · Schur multiplier (H2) = C22 |
| direct product | C127×#12302 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C127 | #1613 |
| commutator, Frattini, cosocle | C2×M16 | C23×C127 |
| prime radical | C2×#41×C127 | D4 |
| LCS | G ▷ C2×M16 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C127 ◁ C2×C4×C127 ◁ C2×M16×C127 ◁ G | |
310 conjugacy classes of subgroups; only the 70 normal subgroups are shown.
This group has 310 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:
(1,2,7,12,8,14,6,13)(3,15,19,36,20,44,18,43)(4,21,23,10,24,39,29,35)(5,25,27,11,28,40,30,38)(9,26,41,17,33,49,32,50)(16,45,47,42,48,63,51,60)(22,37,56,53,57,34,55,58)(31,59,52,54,62,46,61,64)(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);(1,3,8,20)(2,9,14,33)(4,16,24,48)(5,26,28,49)(6,19,7,18)(10,31,35,62)(11,36,38,43)(12,32,13,41)(15,25,44,40)(17,27,50,30)(21,52,39,61)(22,54,57,64)(23,51,29,47)(34,45,37,63)(42,58,60,53)(46,56,59,55);(1,4)(2,10)(3,16)(5,22)(6,29)(7,23)(8,24)(9,31)(11,37)(12,39)(13,21)(14,35)(15,42)(17,46)(18,51)(19,47)(20,48)(25,58)(26,54)(27,56)(28,57)(30,55)(32,61)(33,62)(34,38)(36,63)(40,53)(41,52)(43,45)(44,60)(49,64)(50,59)3175 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(32512, 5290);