Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12310 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1619 |
| commutator | C4×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | C2×#1712 | C2 |
| 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 | |
570 conjugacy classes of subgroups; only the 90 normal subgroups are shown.
This group has 570 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) = 20 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 20 points:
(2,4)(5,6,8,12,17,13,20,19,10,11,18,16,9,7,14,15);(1,2,3,4)(5,7)(6,9)(8,16)(10,13)(11,17)(12,18)(14,15)(19,20);(6,11)(8,14)(9,17)(12,19)(15,16)(18,20)Polycyclic presentation:
2^9|a^2=f,b^2=dfg,d^2=gi,e^2=gh,f^2=gi,g^2=i,h^2=i,b^a=bd,c^a=ce,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=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72213969, #72214250, #72214332, … |
| maximal normal subgroup of | #72213969, #72214250, #72214332, … |
| maximal quotient of | #72214397, #72215012, #72215017, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 44747);