Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12026 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #427 |
| commutator | C2×C8 | C22×C8 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C22 | #1568 |
| prime radical | C23 | C2×#108 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×#97 ◁ #1726 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
210 conjugacy classes of subgroups; only the 106 normal subgroups are shown.
This group has 210 conjugacy classes of subgroups — too many to draw. Only its 106 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 72 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 72 points:
(1,2)(3,6)(4,5)(7,8)(9,10,15,20,16,23,39,54,17,24,14,40,38,22,13,21)(11,25,29,45,30,56,63,46,31,57,28,44,62,43,27,18)(12,32,35,53,36,49,34,51,37,66,41,19,47,50,58,52)(26,42,60,70,61,72,59,69,33,48,64,55,68,71,67,65);(2,5)(6,8)(9,11,13,27,38,62,14,28,17,31,39,63,16,30,15,29)(10,18,40,57,23,45,21,43,24,46,20,25,22,44,54,56)(12,33,34,67,47,61,35,64,37,26,58,59,36,68,41,60)(19,48,50,55,52,71,32,65,53,42,49,70,51,72,66,69);(1,3,4,7)(2,6,5,8)(9,12,13,34,38,47,14,35,17,37,39,58,16,36,15,41)(10,19,20,49,23,52,54,66,24,53,40,50,22,51,21,32)(11,26,27,59,62,68,28,60,31,33,63,67,30,61,29,64)(18,42,25,55,45,72,56,65,46,48,57,70,44,71,43,69)Polycyclic presentation:
2^9|a^2=f,b^2=eg,c^2=d,d^2=gi,e^2=g,f^2=h,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eg,g^a=gi,c^b=ci,f^c=fi104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 42861);