Finite Group DatabasebetaFinite Group Databasebeta| order | 32512 |
| nilpotency class | 5 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C127 · Schur multiplier (H2) = C23 |
| direct product | C127×#13672 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C127 | #2184 |
| commutator, Frattini, cosocle | C2×C16 | C23×C127 |
| prime radical | C127×#3417 | C2 |
| LCS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C127 ◁ C2×C4×C127 ◁ #131×C127 ◁ #504×C127 ◁ G | |
274 conjugacy classes of subgroups; only the 74 normal subgroups are shown.
This group has 274 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) = 255 |
| transitive degree | μt(G) = 16256 |
A minimal faithful permutation representation — 3 generators on 255 points:
(1,2,8,14)(3,15,21,59)(4,22,27,47)(5,28,33,87)(6,34,35,12)(7,36,38,94)(9,39,44,17)(10,26,48,24)(11,49,52,32)(13,53,55,30)(16,60,64,114)(18,58,20,57)(19,69,70,111)(23,74,80,122)(25,46,83,108)(29,54,90,101)(31,92,93,85)(37,50,68,112)(40,97,99,117)(41,65,43,66)(42,88,102,89)(45,103,106,125)(51,91,86,110)(56,67,113,62)(61,71,96,100)(63,118,119,104)(72,109,105,78)(73,81,95,76)(75,115,124,98)(77,84,79,121)(82,116,120,128)(107,126,127,123)(129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255);(1,3)(2,9)(4,16)(5,29)(6,18)(7,37)(8,21)(10,40)(11,50)(12,41)(13,54)(14,44)(15,28)(17,32)(19,30)(20,35)(22,71)(23,75)(24,62)(25,82)(26,67)(27,64)(31,88)(33,90)(34,43)(36,42)(38,68)(39,49)(45,104)(46,107)(47,100)(48,99)(51,111)(52,112)(53,70)(55,101)(56,84)(57,85)(58,92)(59,87)(60,74)(61,78)(63,76)(65,91)(66,110)(69,86)(72,116)(73,115)(77,123)(79,126)(80,124)(81,119)(83,120)(89,93)(94,102)(95,98)(96,109)(97,103)(105,128)(106,118)(108,127)(113,121)(114,122)(117,125);(1,4,8,27)(2,10,14,48)(3,16,21,64)(5,23,33,80)(6,26,35,24)(7,25,38,83)(9,40,44,99)(11,45,52,106)(12,47,34,22)(13,46,55,108)(15,56,59,113)(17,61,39,96)(18,67,20,62)(19,63,70,119)(28,84,87,121)(29,75,90,124)(30,76,53,81)(31,79,93,77)(32,78,49,109)(36,95,94,73)(37,82,68,120)(41,100,43,71)(42,98,102,115)(50,104,112,118)(51,105,86,72)(54,107,101,127)(57,114,58,60)(65,97,66,117)(69,116,111,128)(74,85,122,92)(88,126,89,123)(91,103,110,125)4699 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(32512, 6660);