Finite Group DatabasebetaFinite Group Databasebeta| order | 32512 |
| nilpotency class | 5 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C127 · Schur multiplier (H2) = C2 |
| direct product | C127×#13709 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C127 | #2226 |
| commutator, Frattini, cosocle | C2×C16 | C23×C127 |
| prime radical | #508×C127 | C22 |
| LCS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C127 ◁ C2×C4×C127 ◁ C4×C8×C127 ◁ C4×C16×C127 ◁ G | |
154 conjugacy classes of subgroups; only the 70 normal subgroups are shown.
This group has 154 conjugacy classes of subgroups — too many to draw. Only its 70 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)(3,13,18,45)(4,19)(5,24,26,11)(6,27,30,53)(7,31,16,9)(10,35)(12,41,43,29)(15,46)(17,50,52,40)(20,57)(21,39)(22,54)(23,58)(25,34,61,62)(28,44)(32,56)(33,60)(36,59)(37,63)(38,51)(42,47)(48,55)(49,64)(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);(1,3,5,16,8,18,26,7)(2,9,11,13,14,31,24,45)(4,15,20,47,23,49,22,32)(6,17,25,41,30,52,61,29)(10,33,36,44,39,37,38,55)(12,34,40,27,43,62,50,53)(19,42,54,46,58,56,57,64)(21,48,59,63,35,28,51,60);(1,4)(2,10)(3,15)(5,20)(6,28)(7,32)(8,23)(9,33)(11,36)(12,42)(13,44)(14,39)(16,47)(17,51)(18,49)(19,53)(21,41)(22,26)(24,38)(25,60)(27,58)(29,35)(30,48)(31,37)(34,54)(40,46)(43,56)(45,55)(50,64)(52,59)(57,62)(61,63)4699 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(32512, 6697);