Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12366 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1629 |
| 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 ◁ C22×C4 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
387 conjugacy classes of subgroups; only the 96 normal subgroups are shown.
This group has 387 conjugacy classes of subgroups — too many to draw. Only its 96 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 20 points:
(2,4)(6,10,12,16,13,19,11,18)(7,14)(15,20);(1,2,3,4)(7,15)(8,17)(10,16)(11,12)(14,20)(18,19);(5,6)(7,16)(8,11)(9,13)(10,15)(12,17)(14,18)(19,20)Polycyclic presentation:
2^9|a^2=f,b^2=df,d^2=g,e^2=hi,f^2=g,h^2=i,b^a=bd,c^a=ce,e^a=ei,d^b=dg,e^b=eh,f^b=fg,h^b=hi,d^c=dh,e^c=eh,f^c=fhi,g^c=gi,h^c=hi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72221542, C4×#12366, #72258294, … |
| maximal normal subgroup of | C4×#12366, #72258294, #72258306 |
| maximal quotient of | #72221542 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 44055);