Finite Group DatabasebetaFinite Group Databasebeta| order | 32512 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C127 · Schur multiplier (H2) = C2 |
| direct product | C127×#12464 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C127 | #1675 |
| 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 | |
172 conjugacy classes of subgroups; only the 74 normal subgroups are shown.
This group has 172 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:
(1,2,8,14)(3,15,21,47)(4,22)(5,27,29,12)(6,11,16,42)(7,31,28,13)(9,32,36,18)(10,37)(17,50)(19,45,48,35)(20,34,55,46)(23,41)(24,58)(25,43)(26,38)(30,39)(33,52)(40,49)(44,59)(51,57)(53,61)(54,60)(56,63)(62,64)(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);(1,3,16,48,7,20,5,18,8,21,6,19,28,55,29,32)(2,9,12,34,13,35,11,15,14,36,27,46,31,45,42,47)(4,17,49,64,24,52,23,51,26,54,30,56,25,53,37,59)(10,33,43,62,39,50,38,44,41,61,58,63,40,60,22,57);(1,4)(2,10)(3,17)(5,23)(6,30)(7,24)(8,26)(9,33)(11,38)(12,43)(13,39)(14,41)(15,44)(16,49)(18,51)(19,56)(20,52)(21,54)(22,42)(25,28)(27,58)(29,37)(31,40)(32,59)(34,62)(35,50)(36,61)(45,60)(46,63)(47,57)(48,64)(53,55)4699 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(32512, 5452);