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 ◁ #1451 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
330 conjugacy classes of subgroups; only the 78 normal subgroups are shown.
This group has 330 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) = 128 |
A minimal faithful permutation representation — 3 generators on 40 points:
(2,5,7,8)(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);(1,2,6,5,4,7,3,8)(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);(9,11,15,26)(10,17,21,36)(12,28,13,31)(14,32,29,25)(16,33,34,22)(18,37,19,39)(20,40,38,35)(23,30,24,27)Polycyclic presentation:
2^9|a^2=f,b^2=dfg,c^2=i,d^2=g,e^2=h,f^2=h,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 | #72287160, #72287175, #72287207, … |
| maximal normal subgroup of | #72287160, #72287175, #72287207, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 44230);