Finite Group DatabasebetaFinite Group Databasebeta| order | 32512 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C127 · Schur multiplier (H2) = C23 |
| direct product | C127×#12171 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C127 | #1598 |
| commutator, Frattini, cosocle | C2×M16 | C23×C127 |
| prime radical | #575×C127 | C22 |
| LCS | G ▷ C2×M16 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C127 ◁ C2×C4×C127 ◁ C2×M16×C127 ◁ G | |
332 conjugacy classes of subgroups; only the 70 normal subgroups are shown.
This group has 332 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)(4,18)(5,22)(6,28)(7,12)(8,15)(9,20)(10,27)(11,25)(14,41)(16,48)(17,40)(19,52)(21,34)(23,37)(24,57)(26,38)(29,35)(30,46)(31,47)(32,59)(33,60)(36,45)(39,43)(42,49)(44,62)(50,55)(51,54)(53,61)(56,64)(58,63)(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);(1,3,7,17)(2,8,12,31)(4,14,20,43)(5,23,26,59)(6,29,25,16)(9,30,18,49)(10,35,36,48)(11,37,28,32)(13,38,40,22)(15,45,47,27)(19,53,24,58)(21,56,54,44)(33,62,50,64)(34,63,51,61)(39,52,41,57)(42,60,46,55);(1,4,7,20)(2,9)(3,14,17,43)(5,24,26,19)(6,21,25,54)(8,30)(10,33)(11,34)(12,18)(13,39)(15,46,47,42)(16,44,29,56)(22,57)(23,58,59,53)(27,60,45,55)(28,51)(31,49)(32,61)(35,62)(36,50)(37,63)(38,52)(40,41)(48,64)3175 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(32512, 5159);