Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #138936 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #9259 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C2×#131 | C23 |
| prime radical | C2×#578 | C22 |
| LCS | G ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C2×#131 ◁ G | |
| LECS | G ▷ C2×#131 ▷ C2×C4 ▷ C2 ▷ C1 | |
496 conjugacy classes of subgroups; only the 76 normal subgroups are shown.
This group has 496 conjugacy classes of subgroups — too many to draw. Only its 76 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:
(1,2,7,14,33,15,39,62,8,16,40,64,9,42,6,13)(3,17,22,69,23,88,85,121,24,70,87,112,84,55,21,68)(4,25,28,54,43,61,27,65,29,92,38,11,30,52,96,53)(5,12,35,57,73,58,34,45,36,59,41,63,37,103,18,71)(10,44,48,82,67,86,47,99,49,102,60,20,50,81,110,31)(19,74,77,114,78,120,76,106,79,122,83,66,89,119,97,117)(26,91,93,51,32,100,101,56,94,105,98,115,95,116,72,118)(46,80,90,104,113,125,111,126,108,127,128,123,109,75,107,124);(1,3,18,44,8,24,34,102)(2,10,45,68,16,49,71,121)(4,19,72,104,29,79,101,123)(5,31,7,21,36,99,40,85)(6,22,73,20,39,87,37,82)(9,23,41,86,33,84,35,81)(11,46,105,120,54,108,91,119)(12,55,64,110,59,88,14,47)(13,60,103,17,62,48,58,70)(15,50,63,112,42,67,57,69)(25,90,56,117,92,128,118,106)(26,80,28,97,94,127,38,76)(27,83,95,125,96,77,32,75)(30,78,98,126,43,89,93,124)(51,114,61,107,115,66,52,111)(53,109,116,74,65,113,100,122);(1,4)(2,11)(3,19)(5,32)(6,38)(7,27)(8,29)(9,43)(10,46)(12,56)(13,61)(14,25)(15,65)(16,54)(17,66)(18,72)(20,80)(21,83)(22,76)(23,89)(24,79)(26,73)(28,39)(30,33)(31,75)(34,101)(35,98)(36,95)(37,94)(40,96)(41,93)(42,53)(44,104)(45,105)(47,90)(48,111)(49,108)(50,113)(51,58)(52,62)(55,117)(57,116)(59,118)(60,107)(63,100)(64,92)(67,109)(68,120)(69,74)(70,114)(71,91)(77,85)(78,84)(81,126)(82,127)(86,124)(87,97)(88,106)(99,125)(102,123)(103,115)(110,128)(112,122)(119,121)Polycyclic presentation:
2^9|a^2=f,b^2=dfh,d^2=gh,e^2=ghi,f^2=gh,h^2=i,b^a=bd,c^a=ce,d^a=dg,g^a=gi,h^a=hi,d^b=dh,e^b=egh,f^b=fhi,g^b=gi,d^c=dgh,e^c=egh,f^c=fgh,h^c=hi,f^d=fi47 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 46132);