Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #137558 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #9061 |
| commutator | C4×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | #19660 | C2 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
459 conjugacy classes of subgroups; only the 88 normal subgroups are shown.
This group has 459 conjugacy classes of subgroups — too many to draw. Only its 88 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2,6,11,7,12,22,30,5,13,4,19,8,10,21,28)(3,14,17,25,16,29,23,26,18,31,15,27,24,32,20,9);(1,3)(2,9,13,26)(4,20)(5,18)(6,23)(7,24)(8,16)(10,25,12,27)(11,29,19,32)(14,28,31,30)(15,21)(17,22);(2,10)(4,21)(6,22)(7,8)(9,25)(12,13)(14,31)(15,20)(16,24)(17,23)(26,27)(28,30)Polycyclic presentation:
2^9|a^2=f,b^2=dfg,d^2=g,e^2=gh,f^2=g,g^2=i,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,g^b=gi,d^c=dg,e^c=egh,f^c=fg,g^c=gi,h^c=hi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72219594, #72219597, #72219600, … |
| maximal normal subgroup of | #72219594, #72219597, #72219600, … |
| maximal quotient of | #72226843 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 44754);