Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #136374 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1864 |
| commutator | C4×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | C23 | #450 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ C8×C4⋊C4 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
348 conjugacy classes of subgroups; only the 96 normal subgroups are shown.
This group has 348 conjugacy classes of subgroups — too many to draw. Only its 96 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 40 points:
(1,2,4,7)(3,5,8,6)(9,10,15,21,16,23,36,22)(11,24,27,40,12,28,31,37)(13,32,33,19,34,20,14,35)(17,30,38,25,39,26,18,29);(2,5)(3,8)(6,7)(9,11,16,12)(10,17)(13,26,34,30)(14,29,33,25)(15,31,36,27)(18,21)(19,37)(20,28)(22,38)(23,39)(24,32)(35,40);(9,12,16,11)(10,18,23,38)(13,30,34,26)(14,25,33,29)(15,27,36,31)(17,21,39,22)(19,28,35,24)(20,40,32,37)Polycyclic presentation:
2^9|a^2=f,b^2=fg,c^2=i,d^2=g,e^2=hi,f^2=i,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ehi,g^a=gi,h^a=hi,d^b=dg,e^b=eh,f^b=fi,g^b=gi,h^b=hi,d^c=dh,e^c=eh,f^c=fi,g^c=gi,h^c=hi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 43572);