Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #152644 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #13059 |
| commutator | C2×C16 | C22×C4 |
| Frattini, cosocle | C22×C16 | C23 |
| prime radical | C2×#41 | D8 |
| LCS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ C2×#131 ◁ C2×#504 ◁ G | |
| LECS | G ▷ C22×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
302 conjugacy classes of subgroups; only the 92 normal subgroups are shown.
This group has 302 conjugacy classes of subgroups — too many to draw. Only its 92 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 128 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 128 points:
(1,2)(3,15)(4,22,24,10)(5,27)(6,33)(7,36)(8,12)(9,18)(11,30)(13,29)(14,34)(16,46)(17,49,58,93)(19,53)(20,68)(21,69)(23,73,28,81)(25,78,76,89)(26,44,77,72)(31,88)(32,37)(35,90)(38,57,70,61)(39,96)(40,98)(41,63)(42,55,45,83)(43,74,71,82)(47,56)(48,86)(50,64)(51,62)(52,67)(54,105)(59,113,111,128)(60,114,112,103)(65,91)(66,119)(75,123,84,122)(79,85)(80,87)(92,118)(94,126,120,127)(95,117,121,110)(97,107)(99,125,100,108)(101,116,104,106)(102,115,124,109);(1,3,16,54,6,20,31,65,34,67,85,96,86,118,5,18,8,19,30,64,35,66,87,98,7,21,56,107,29,62,32,63)(2,9,37,92,13,39,47,52,36,91,80,68,90,105,11,15,12,41,27,51,48,97,79,69,14,40,88,119,33,50,46,53)(4,17,55,106,25,59,84,117,77,112,125,127,82,115,28,61,24,58,83,116,76,111,75,110,26,60,108,126,74,109,23,57)(10,38,81,124,43,94,100,103,72,121,122,113,78,101,45,49,22,70,73,102,71,120,99,114,44,95,123,128,89,104,42,93);(1,4)(2,10)(3,17)(5,28)(6,25)(7,26)(8,24)(9,38)(11,45)(12,22)(13,43)(14,44)(15,49)(16,55)(18,61)(19,58)(20,59)(21,60)(23,32)(27,73)(29,74)(30,83)(31,84)(33,89)(34,77)(35,76)(36,72)(37,81)(39,94)(40,95)(41,70)(42,46)(47,100)(48,71)(50,104)(51,102)(52,103)(53,93)(54,106)(56,108)(57,63)(62,109)(64,116)(65,117)(66,111)(67,112)(68,113)(69,114)(75,87)(78,90)(79,99)(80,122)(82,86)(85,125)(88,123)(91,121)(92,124)(96,127)(97,120)(98,110)(101,105)(107,126)(115,118)(119,128)Polycyclic presentation:
2^9|a^2=f,b^2=dfgi,d^2=gh,g^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ei,g^a=gh,h^a=hi,e^b=ei,d^c=di,f^c=fi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 59841);