Finite Group DatabasebetaFinite Group Databasebeta| order | 32512 |
| nilpotency class | 5 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C127 · Schur multiplier (H2) = C22 |
| direct product | C127×#13690 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C127 | C2×D16 |
| commutator, Frattini, cosocle | C2×C16 | C23×C127 |
| prime radical | C2×D16×C127 | C22 |
| LCS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C127 ◁ C42×C127 ◁ C4×C8×C127 ◁ C4×C16×C127 ◁ G | |
204 conjugacy classes of subgroups; only the 74 normal subgroups are shown.
This group has 204 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) = 195 |
| transitive degree | μt(G) = 16256 |
A minimal faithful permutation representation — 3 generators on 195 points:
(2,4)(5,6)(7,17)(8,12)(9,25)(10,30)(11,32)(13,19)(14,28)(15,27)(16,24)(18,43)(20,50)(21,52)(22,44)(23,55)(26,41)(29,62)(31,39)(33,45)(34,46)(35,48)(36,51)(37,60)(38,59)(40,67)(42,49)(47,68)(53,66)(54,58)(56,65)(57,61)(63,64)(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);(1,2,3,4)(6,12)(13,34)(14,35)(15,36)(16,32)(17,43)(25,44)(30,55)(37,64)(38,42)(39,65)(40,57)(41,66)(45,52)(50,68)(58,62);(5,7,8,18)(6,13,12,34)(9,26,22,53)(10,31,23,56)(11,33,24,21)(14,39,35,65)(15,41,36,66)(16,42,32,38)(17,44,43,25)(19,48,46,28)(20,51,47,27)(29,63,54,60)(30,37,55,64)(40,68,57,50)(45,58,52,62)(49,61,59,67)6223 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(32512, 6678);