Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #141445 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #9783 |
| 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 | |
315 conjugacy classes of subgroups; only the 76 normal subgroups are shown.
This group has 315 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:
(1,2,7,14,9,16,8,15)(3,17,22,62,24,52,23,63)(4,25,29,80,30,81,39,99)(5,31,34,42,18,58,40,12)(6,35,26,13,38,48,37,56)(10,41,45,64,46,77,59,20)(11,47,50,28,51,89,57,90)(19,66,70,118,71,119,76,125)(21,74,67,113,36,98,75,61)(27,83,86,79,32,93,87,114)(33,95,84,111,65,54,82,92)(43,102,106,124,108,128,107,121)(44,96,97,120,55,73,103,101)(49,85,91,110,53,94,112,127)(60,109,115,69,104,123,116,88)(68,78,100,117,72,105,122,126);(1,3,18,64,8,22,40,41,9,24,5,20,7,23,34,77)(2,10,42,17,14,45,31,63,16,46,12,52,15,59,58,62)(4,19,27,68,29,76,87,100,30,71,32,72,39,70,86,122)(6,36,65,73,37,75,82,120,38,21,33,96,26,67,84,101)(11,43,49,60,57,107,91,115,51,108,53,104,50,106,112,116)(13,55,95,113,48,103,92,74,56,44,54,61,35,97,111,98)(25,78,79,119,99,126,93,125,81,105,114,66,80,117,83,118)(28,88,85,124,89,123,127,128,90,69,94,121,47,109,110,102);(1,4,26,47,8,29,6,28,9,30,37,89,7,39,38,90)(2,11,48,99,14,57,56,81,16,51,35,80,15,50,13,25)(3,19,67,109,22,76,36,88,24,71,75,123,23,70,21,69)(5,32,82,127,34,86,33,94,18,27,84,110,40,87,65,85)(10,43,103,126,45,107,44,105,46,108,97,117,59,106,55,78)(12,53,111,83,58,112,95,79,42,49,92,93,31,91,54,114)(17,60,74,125,63,115,61,66,52,104,98,118,62,116,113,119)(20,72,120,128,77,122,96,121,64,68,101,102,41,100,73,124)Polycyclic presentation:
2^9|a^2=f,b^2=dfg,c^2=egi,d^2=gh,e^2=gh,f^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=egh,h^a=hi,d^b=dh,e^b=ei,f^b=fhi,g^b=gi,h^b=hi,d^c=di,f^c=fi,f^d=fi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 48645);