Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #143319 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1629 |
| commutator, Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | #28502 | C2 |
| LCS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
513 conjugacy classes of subgroups; only the 82 normal subgroups are shown.
This group has 513 conjugacy classes of subgroups — too many to draw. Only its 82 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:
(6,10,12,20)(7,14)(8,17)(11,18,13,15)(16,19);(1,2)(3,4)(7,14)(8,17)(10,18)(11,13)(15,20)(16,19);(2,4)(5,6)(7,15)(8,11)(9,12)(10,19)(13,17)(14,20)(16,18)Polycyclic presentation:
2^9|a^2=g,d^2=g,e^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,h^a=hi,c^b=cf,d^b=dg,e^b=eh,h^b=hi,d^c=dh,e^c=ehi,g^c=gi,h^c=hi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72214352, #72214605, #72214744, … |
| maximal normal subgroup of | #72214352, #72214744, #72214756, … |
| maximal quotient of | #72214381, #72214409, #72214416, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 50515);