Finite Group DatabasebetaFinite Group Databasebeta| order | 32512 |
| nilpotency class | 6 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C127 · Schur multiplier (H2) = C22 |
| direct product | C2×C127×SD128 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C127 | D32 |
| commutator, Frattini, cosocle | C32 | C23×C127 |
| prime radical | C2×D32×C127 | C2 |
| LCS | G ▷ C32 ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C127 ◁ C2×C4×C127 ◁ C2×C8×C127 ◁ C2×C16×C127 ◁ C2×C32×C127 ◁ G | |
134 conjugacy classes of subgroups; only the 62 normal subgroups are shown.
This group has 134 conjugacy classes of subgroups — too many to draw. Only its 62 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 193 |
| transitive degree | μt(G) = 16256 |
A minimal faithful permutation representation — 3 generators on 193 points:
(1,2)(67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193);(3,4,9,15)(5,10,20,32)(6,21,24,55)(7,25,27,58)(8,28,26,14)(11,33,36,16)(12,37,39,17)(13,40,42,22)(18,41,51,66)(19,43,50,31)(23,56,47,54)(29,60,62,45)(30,63,64,44)(34,65,61,49)(35,46,59,52)(38,48,53,57);(4,10)(5,16)(6,17)(7,22)(8,26)(11,12)(13,41)(14,43)(15,32)(18,49)(19,52)(20,33)(21,29)(23,57)(24,37)(25,30)(27,40)(28,31)(34,54)(35,53)(36,39)(38,59)(42,66)(44,45)(46,50)(47,48)(51,65)(55,62)(56,61)(58,64)(60,63)8890 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(32512, 6727);