Finite Group DatabasebetaFinite Group Databasebeta| order | 32512 |
| nilpotency class | 5 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C127 · Schur multiplier (H2) = C23 |
| direct product | C127×#13666 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C127 | C2×D16 |
| commutator, Frattini, cosocle | C2×C16 | C23×C127 |
| prime radical | C2×#508×C127 | C2 |
| LCS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C127 ◁ C22×C4×C127 ◁ C22×C8×C127 ◁ C22×C16×C127 ◁ G | |
288 conjugacy classes of subgroups; only the 78 normal subgroups are shown.
This group has 288 conjugacy classes of subgroups — too many to draw. Only its 78 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 195 |
| transitive degree | μt(G) = 16256 |
A minimal faithful permutation representation — 3 generators on 195 points:
(1,2)(3,4)(5,6,11,17)(7,18,23,49)(8,13,26,16)(9,27,30,61)(10,31,33,39)(12,34,37,20)(14,40,43,29)(15,44,46,25)(19,48,22,55)(21,53,54,67)(24,56,59,60)(28,45,63,68)(32,41,51,66)(35,52,47,50)(36,57,64,62)(38,65,42,58)(69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195);(5,7)(6,12)(8,19)(9,28)(10,32)(11,23)(13,35)(14,41)(15,45)(16,47)(17,37)(18,27)(20,29)(21,25)(22,26)(24,57)(30,63)(31,36)(33,51)(34,40)(38,53)(39,64)(42,67)(43,66)(44,54)(46,68)(48,56)(49,61)(50,58)(52,65)(55,60)(59,62);(2,4)(5,8,11,26)(6,13,17,16)(7,19,23,22)(9,24,30,59)(10,25,33,44)(12,35,37,47)(14,38,43,42)(15,39,46,31)(18,48,49,55)(20,50,34,52)(21,51,54,32)(27,56,61,60)(28,57,63,62)(29,58,40,65)(36,45,64,68)(41,53,66,67)6223 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(32512, 6654);