Finite Group DatabasebetaFinite Group Databasebeta| order | 32512 |
| nilpotency class | 5 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C127 · Schur multiplier (H2) = C23 |
| direct product | C127×#13671 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C127 | #2184 |
| commutator, Frattini, cosocle | C2×C16 | C23×C127 |
| prime radical | C127×#3415 | C2 |
| LCS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C127 ◁ C2×C4×C127 ◁ #131×C127 ◁ #504×C127 ◁ G | |
284 conjugacy classes of subgroups; only the 74 normal subgroups are shown.
This group has 284 conjugacy classes of subgroups — too many to draw. Only its 74 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:
(1,2,8,15)(3,16,20,45)(4,21,27,54)(5,14,7,12)(6,30,32,46)(9,33,37,17)(10,28,39,18)(11,40,44,25)(13,29,48,23)(19,51,52,38)(22,47,57,64)(24,34,26,53)(31,41,61,63)(35,50,49,56)(36,60,62,55)(42,59,43,58)(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)(4,22)(5,18)(6,31)(7,28)(8,20)(10,34)(11,41)(12,35)(13,47)(14,49)(15,37)(16,21)(17,25)(19,23)(24,55)(26,60)(27,57)(29,52)(30,36)(32,61)(33,40)(38,43)(39,53)(42,51)(44,63)(45,54)(46,62)(48,64)(50,59)(56,58);(1,3)(2,10)(4,17)(5,28)(6,19)(7,18)(8,20)(9,34)(11,36)(12,45)(13,38)(14,16)(15,39)(21,49)(22,25)(23,31)(24,56)(26,50)(27,33)(29,61)(30,41)(32,52)(35,54)(37,53)(40,57)(42,64)(43,47)(44,62)(46,63)(48,51)(55,58)(59,60)4699 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(32512, 6659);