Finite Group DatabasebetaFinite Group Databasebeta| order | 32512 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C127 · Schur multiplier (H2) = C2 |
| direct product | C127×#12463 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C127 | #1675 |
| commutator, Frattini, cosocle | C4×C8 | C23×C127 |
| prime radical | #135×C127 | D4 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C127 ◁ C2×C4×C127 ◁ C4×C8×C127 ◁ G | |
172 conjugacy classes of subgroups; only the 74 normal subgroups are shown.
This group has 172 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:
(1,2,8,14)(3,15,20,47)(4,21)(5,26,28,12)(6,11,30,42)(7,31,27,13)(9,32,36,17)(10,37)(16,48)(18,45,55,35)(19,34,53,46)(22,41)(23,57)(24,43)(25,38)(29,39)(33,50)(40,58)(44,59)(49,56)(51,61)(52,60)(54,63)(62,64)(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);(1,3,6,18,7,19,28,32,8,20,30,55,27,53,5,17)(2,9,26,46,13,35,42,47,14,36,12,34,31,45,11,15)(4,16,29,54,23,50,37,59,25,52,58,64,24,51,22,49)(10,33,57,63,39,48,21,56,41,61,43,62,40,60,38,44);(1,4)(2,10)(3,16)(5,22)(6,29)(7,23)(8,25)(9,33)(11,38)(12,43)(13,39)(14,41)(15,44)(17,49)(18,54)(19,50)(20,52)(21,42)(24,27)(26,57)(28,37)(30,58)(31,40)(32,59)(34,62)(35,48)(36,61)(45,60)(46,63)(47,56)(51,53)(55,64)4699 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(32512, 5451);