Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12430 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #466 |
| commutator | C4×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C22 | #1893 |
| prime radical | #3147 | C22 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ C2×C8×Q8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
349 conjugacy classes of subgroups; only the 102 normal subgroups are shown.
This group has 349 conjugacy classes of subgroups — too many to draw. Only its 102 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:
(1,2,5,8,6,3,4,7)(9,10,13,18,15,19,14,29)(11,20,22,35,24,36,23,39)(12,25,26,31,28,40,27,17)(16,30,32,38,33,37,34,21);(1,3,6,2)(4,7,5,8)(9,11,12,21,13,22,26,30,15,24,28,38,14,23,27,37)(10,16,31,36,29,34,25,35,19,33,17,20,18,32,40,39);(1,2,6,3)(4,8,5,7)Polycyclic presentation:
2^9|a^2=f,b^2=dfgh,c^2=h,d^2=ghi,e^2=h,f^2=hi,g^2=i,b^a=bd,c^a=ce,d^a=dg,g^a=gi,d^b=dh,e^b=eh,f^b=fh,d^c=dh,e^c=eh,f^c=fh98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 45869);