Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #137493 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1879 |
| commutator | C4×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | #33719 | C2 |
| 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 | |
473 conjugacy classes of subgroups; only the 90 normal subgroups are shown.
This group has 473 conjugacy classes of subgroups — too many to draw. Only its 90 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) = 64 |
A minimal faithful permutation representation — 3 generators on 40 points:
(1,2)(3,5)(4,7)(6,8)(9,10,13,18,14,20,30,34,15,21,12,19,29,38,27,16)(11,22,24,35,25,37,31,36,26,33,23,40,32,39,28,17);(10,16,21,34)(12,27)(13,30)(14,29)(17,33,36,22)(18,20,19,38)(23,28)(24,31)(25,32)(35,37,40,39);(2,5)(3,8)(6,7)(9,11)(10,17)(12,28)(13,31)(14,32)(15,26)(16,33)(18,37)(19,39)(20,40)(21,36)(22,34)(23,27)(24,30)(25,29)(35,38)Polycyclic presentation:
2^9|a^2=f,b^2=dfg,d^2=g,e^2=ghi,f^2=g,g^2=i,b^a=bd,c^a=ce,d^a=di,e^a=eh,d^b=dg,e^b=eg,f^b=fg,g^b=gi,d^c=dg,e^c=egh,f^c=fg,g^c=gi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 44739);