Finite Group DatabasebetaFinite Group Databasebeta| order | 32512 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C127 · Schur multiplier (H2) = C24 |
| direct product | C127×#13096 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C127 | #1898 |
| commutator | C22×C4 | C22×C4×C127 |
| Frattini, cosocle | C2×#41 | C23×C127 |
| prime radical | C2×2−1+4×C127 | C22 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C127 ◁ C2×C4×C127 ◁ C2×2−1+4×C127 ◁ G | |
624 conjugacy classes of subgroups.
| minimal degree | μ(G) = 191 |
| transitive degree | μt(G) = 8128 |
A minimal faithful permutation representation — 3 generators on 191 points:
(1,2)(3,13,18,42)(4,19,22,38)(5,23,14,10)(6,11)(7,12)(8,16)(9,21,34,28)(15,44)(17,41,27,58)(20,53)(24,32)(25,52)(26,55,43,36)(29,39)(30,59,62,64)(31,49)(33,54)(35,57)(37,56)(40,50)(45,51,48,61)(46,60)(47,63)(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);(1,3,14,16,7,18,5,24)(2,8,23,42,12,32,10,13)(4,15,20,48,22,47,25,45)(6,27,43,56,29,17,26,49)(9,30,33,60,34,62,35,40)(11,37,55,41,39,31,36,58)(19,51,52,44,38,61,53,63)(21,50,54,64,28,46,57,59);(1,4)(2,9)(3,15)(5,25)(6,28)(7,22)(8,30)(10,35)(11,38)(12,34)(13,40)(14,20)(16,48)(17,50)(18,47)(19,39)(21,29)(23,33)(24,45)(26,54)(27,46)(31,51)(32,62)(36,52)(37,61)(41,63)(42,60)(43,57)(44,58)(49,64)(53,55)(56,59)4318 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(32512, 6084);