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 | C2×#135 | D4 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ #1770 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
374 conjugacy classes of subgroups; only the 98 normal subgroups are shown.
This group has 374 conjugacy classes of subgroups — too many to draw. Only its 98 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:
(1,2,4,6,5,8,3,7)(9,10,15,22,17,24,13,35,18,25,14,36,16,23,27,21)(11,26,29,39,31,37,12,33,32,40,34,19,30,20,28,38);(2,6)(3,4)(7,8)(9,11,17,30,18,32,16,31)(10,19,23,38,25,39,24,33)(12,27,29,15,28,13,34,14)(20,36,26,35,37,22,40,21);(9,12)(10,20)(11,27)(13,32)(14,31)(15,30)(16,34)(17,29)(18,28)(19,36)(21,33)(22,39)(23,26)(24,40)(25,37)(35,38)Polycyclic presentation:
2^9|a^2=f,b^2=dfhi,d^2=g,e^2=h,f^2=gi,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 | #72309455, #72309460, #72309463, … |
| maximal normal subgroup of | #72309455, #72309460, #72309463, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 44156);