Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #136906 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
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 | #3169 | C22 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ #1451 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
329 conjugacy classes of subgroups; only the 78 normal subgroups are shown.
This group has 329 conjugacy classes of subgroups — too many to draw. Only its 78 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) = 256 |
A minimal faithful permutation representation — 3 generators on 40 points:
(2,5,7,8)(9,10,12,21,14,22,13,17,16,24,33,19,15,23,32,20)(11,25,26,18,28,36,27,37,30,40,34,39,29,35,31,38);(1,2,6,5,4,7,3,8)(10,17,24,20)(12,13)(14,15)(18,35,39,36)(19,22,21,23)(25,37,40,38)(26,27)(28,29)(31,34)(32,33);(9,11,16,30)(10,18,24,39)(12,31,33,27)(13,34,32,26)(14,29,15,28)(17,35,20,36)(19,40,21,25)(22,38,23,37)Polycyclic presentation:
2^9|a^2=f,b^2=dfgi,c^2=i,d^2=g,e^2=h,f^2=hi,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=di,e^a=ei,d^b=dg,e^b=eh,f^b=fg,g^b=gi,h^b=hi,d^c=dh,e^c=ehi,f^c=fh,g^c=gi,h^c=hi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72287310, #72287327, #72287353, … |
| maximal normal subgroup of | #72287310, #72287327, #72287353, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 44236);