Finite Group DatabasebetaFinite Group Databasebeta| order | 32512 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C127 · Schur multiplier (H2) = C22 |
| direct product | C127×#12437 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C127 | #1667 |
| 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 | |
214 conjugacy classes of subgroups; only the 74 normal subgroups are shown.
This group has 214 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:
(1,2,9,16)(3,17,23,47)(4,24,27,55)(5,13,29,41)(6,12,28,42)(7,15,8,14)(10,32,37,19)(11,38,40,25)(18,48,51,59)(20,36,52,35)(21,46,22,34)(26,56,31,43)(30,44,57,39)(33,50,60,54)(45,58,62,49)(53,63,64,61)(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);(1,3,6,20,7,21,29,32,9,23,28,52,8,22,5,19)(2,10,13,34,14,35,42,47,16,37,41,46,15,36,12,17)(4,18,30,53,26,50,38,58,27,51,57,64,31,54,25,49)(11,33,43,61,39,59,55,62,40,60,56,63,44,48,24,45);(1,4)(2,11)(3,18)(5,25)(6,30)(7,26)(8,31)(9,27)(10,33)(12,24)(13,43)(14,39)(15,44)(16,40)(17,45)(19,49)(20,53)(21,50)(22,54)(23,51)(28,57)(29,38)(32,58)(34,61)(35,59)(36,48)(37,60)(41,56)(42,55)(46,63)(47,62)(52,64)4699 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(32512, 5425);