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 | |
499 conjugacy classes of subgroups; only the 76 normal subgroups are shown.
This group has 499 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:
(1,2,6,12,36,14,60,59,8,13,38,58,7,39,37,103)(3,15,19,63,21,81,80,106,20,64,79,114,74,51,34,101)(4,22,24,50,40,86,102,49,26,85,93,61,25,57,35,10)(5,27,31,53,89,95,16,42,33,11,30,54,32,55,67,107)(9,41,45,77,104,75,56,97,47,65,44,18,46,76,115,28)(17,68,70,108,72,128,127,120,71,123,126,119,82,121,78,62)(23,87,91,94,100,118,66,117,92,84,90,52,29,98,99,48)(43,69,110,125,116,88,83,105,112,96,109,124,111,73,113,122);(1,3,16,65)(2,9,42,106)(4,17,66,122)(5,28,6,34)(7,21,31,76)(8,20,67,41)(10,43,98,123)(11,51,58,115)(12,56,27,81)(13,47,107,101)(14,46,54,63)(15,59,45,95)(18,60,79,89)(19,32,77,37)(22,83,117,120)(23,88,24,78)(25,72,91,125)(26,71,99,105)(29,96,35,70)(30,75,36,74)(33,97,38,80)(39,104,53,114)(40,82,90,124)(44,55,64,103)(48,62,85,113)(49,112,118,68)(50,111,87,128)(52,108,86,110)(57,109,94,119)(61,116,84,121)(69,102,126,100)(73,93,127,92);(1,4,8,26)(2,10,13,49)(3,17,20,71)(5,29,33,100)(6,35,38,102)(7,40,36,25)(9,43,47,112)(11,52,27,94)(12,57,58,86)(14,61,39,50)(15,62,64,120)(16,66,67,99)(18,73,77,88)(19,78,79,127)(21,82,74,72)(22,59,85,103)(23,89,92,32)(24,60,93,37)(28,96,97,69)(30,91,31,90)(34,70,80,126)(41,105,65,122)(42,98,107,118)(44,83,45,113)(46,116,104,111)(48,55,117,95)(51,108,81,119)(53,87,54,84)(56,109,115,110)(63,121,114,128)(68,106,123,101)(75,125,76,124)Polycyclic presentation:
2^9|a^2=f,b^2=dfhi,c^2=i,d^2=gh,e^2=ghi,f^2=ghi,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, 46139);