Finite Group DatabasebetaFinite Group Databasebeta| order | 32512 |
| nilpotency class | 5 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C127 · Schur multiplier (H2) = C23 |
| direct product | C127×#13615 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C127 | D16 |
| commutator | C16 | C22×C4×C127 |
| Frattini, cosocle | C2×C16 | C23×C127 |
| socle | C22×C127 | C2×D16 |
| prime radical | C2×#508×C127 | C2 |
| LCS | G ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C127 ◁ C22×C4×C127 ◁ C22×C8×C127 ◁ C22×C16×C127 ◁ G | |
298 conjugacy classes of subgroups; only the 126 normal subgroups are shown.
This group has 298 conjugacy classes of subgroups — too many to draw. Only its 126 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)(7,18)(8,13)(9,27)(10,31)(11,17)(12,20)(14,29)(15,25)(16,26)(19,45)(21,51)(22,53)(23,46)(24,54)(28,42)(30,60)(32,39)(33,38)(34,47)(35,61)(36,50)(37,56)(40,66)(41,63)(43,58)(44,49)(48,65)(52,67)(55,64)(57,59)(62,68)(69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195);(5,7)(6,12)(8,19)(9,28)(10,32)(11,23)(13,34)(14,39)(15,42)(16,44)(17,36)(18,27)(20,29)(21,25)(22,26)(24,55)(30,62)(31,35)(33,48)(37,51)(38,64)(40,67)(41,65)(43,68)(45,54)(46,60)(47,56)(49,66)(50,63)(52,58)(53,59)(57,61);(5,8,11,26)(6,13,17,16)(7,19,23,22)(9,24,30,57)(10,25,33,58)(12,34,36,44)(14,37,41,40)(15,38,43,31)(18,45,46,53)(20,47,50,49)(21,48,52,32)(27,54,60,59)(28,55,62,61)(29,56,63,66)(35,42,64,68)(39,51,65,67)9652 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(32512, 6603);