Finite Group DatabasebetaFinite Group Databasebeta| order | 32512 |
| nilpotency class | 5 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C127 · Schur multiplier (H2) = C4 |
| direct product | C127×#13724 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C22×C127 | C2×D16 |
| commutator, Frattini, cosocle | C2×C16 | C23×C127 |
| LCS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C127 ◁ C42×C127 ◁ C4×C8×C127 ◁ C4×C16×C127 ◁ G | |
174 conjugacy classes of subgroups; only the 78 normal subgroups are shown.
This group has 174 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) = 199 |
| transitive degree | μt(G) = 32512 |
A minimal faithful permutation representation — 3 generators on 199 points:
(1,2)(3,6)(4,5)(7,8)(9,10,17,24)(11,19,29,49)(12,18,32,46)(13,21,35,42)(14,20,25,51)(15,38,40,67)(16,41,39,23)(22,54,55,33)(26,47,56,71)(27,60,62,72)(28,63,61,48)(30,43,64,68)(31,65,58,45)(34,66,36,53)(37,52,57,50)(44,69,70,59)(73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199);(2,5)(6,8)(9,11,25,56,15,27,34,31,39,61,37,59,54,64,13,12,17,29,14,26,40,62,36,58,16,28,57,69,33,30,35,32)(10,18,42,68,22,44,50,48,41,65,53,60,67,71,20,19,24,46,21,43,55,70,52,63,23,45,66,72,38,47,51,49);(1,3,4,7)(2,6,5,8)(9,12,13,30,33,59,37,28,16,31,34,62,40,56,25,29,17,32,35,64,54,69,57,61,39,58,36,27,15,26,14,11)(10,19,20,47,38,60,53,45,23,48,50,70,55,68,42,46,24,49,51,71,67,72,66,65,41,63,52,44,22,43,21,18)5842 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(32512, 6712);