Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #141499 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #9785 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C2×#131 | C23 |
| prime radical | C2×#41 | C2×D4 |
| LCS | G ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C2×#131 ◁ G | |
| LECS | G ▷ C2×#131 ▷ C2×C4 ▷ C2 ▷ C1 | |
309 conjugacy classes of subgroups; only the 72 normal subgroups are shown.
This group has 309 conjugacy classes of subgroups — too many to draw. Only its 72 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)(3,15,20,46)(4,22,26,75)(5,28,32,91)(6,33)(7,13)(8,14)(9,18)(10,25,44,87)(11,16,49,61)(12,50)(17,62)(19,70,34,99)(21,57,37,102)(23,77)(24,81)(27,76,83,127)(29,92)(30,54)(31,95,82,48)(35,52)(36,98)(38,103,106,125)(39,107)(40,68)(41,69)(42,109,112,128)(43,113)(45,88,116,120)(47,118)(51,121)(53,58)(55,100)(56,101,117,72)(59,74)(60,108,71,90)(63,122,66,105)(64,104)(65,124)(67,123)(73,93,111,78)(79,110)(80,126)(84,89)(85,119)(86,114)(94,115)(96,97);(1,3,16,58,30,21,5,18,8,20,61,68,7,37,32,69)(2,9,28,57,13,53,49,15,14,41,91,102,54,40,11,46)(4,23,59,66,83,65,29,93,26,80,85,63,27,17,24,78)(6,34,60,107,50,101,31,96,36,19,71,121,35,72,82,67)(10,42,89,104,45,38,43,114,44,112,115,55,116,106,47,110)(12,51,90,70,33,97,95,117,52,39,108,99,98,123,48,56)(22,73,92,62,76,122,74,126,75,111,81,124,127,105,119,77)(25,86,118,125,120,64,94,128,87,79,113,103,88,100,84,109);(1,4,6,25,30,83,50,120,8,26,36,87,7,27,35,88)(2,10,12,22,13,45,33,76,14,44,52,75,54,116,98,127)(3,17,19,64,21,23,72,79,20,65,34,100,37,80,101,86)(5,29,31,94,61,85,71,113,32,24,82,84,16,59,60,118)(9,38,39,105,53,112,123,73,41,106,51,122,40,42,97,111)(11,47,48,119,28,89,90,92,49,43,95,74,91,115,108,81)(15,55,56,62,102,110,70,126,46,104,117,124,57,114,99,77)(18,66,67,103,68,93,107,109,69,63,96,125,58,78,121,128)Polycyclic presentation:
2^9|a^2=f,b^2=dfgi,c^2=e,d^2=gh,e^2=gh,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=egh,h^a=hi,c^b=cg,d^b=dh,e^b=ei,f^b=fhi,g^b=gi,h^b=hi,d^c=di,f^c=fi,f^d=fi44 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 48697);