Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #137493 |
| 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 | #3147 | C22 |
| 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 | |
361 conjugacy classes of subgroups; only the 90 normal subgroups are shown.
This group has 361 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) = 128 |
A minimal faithful permutation representation — 3 generators on 40 points:
(1,2,5,8,6,3,4,7)(9,10,14,18,15,20,12,16)(11,21,24,37,25,38,22,35)(13,27,29,36,30,32,26,19)(17,28,33,40,34,39,31,23);(1,3,6,2)(4,7,5,8)(10,16,20,18)(17,31,34,33)(19,32,36,27)(21,35,38,37);(1,2,6,3)(4,8,5,7)(9,11)(10,17)(12,22)(13,28)(14,24)(15,25)(16,31)(18,33)(19,35)(20,34)(21,27)(23,26)(29,40)(30,39)(32,38)(36,37)Polycyclic presentation:
2^9|a^2=f,b^2=dfg,c^2=g,d^2=gi,e^2=ghi,f^2=gi,h^2=i,b^a=bd,c^a=ce,d^a=di,e^a=eh,h^a=hi,d^b=dg,e^b=eg,f^b=fg,d^c=dg,e^c=egh,f^c=fg,h^c=hi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 44771);