Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11992 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1555 |
| commutator | C42 | C2×C42 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | C24 | #100 |
| LCS | G ▷ C42 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C4×M16 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
440 conjugacy classes of subgroups; only the 146 normal subgroups are shown.
This group has 440 conjugacy classes of subgroups — too many to draw. Only its 146 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:
(1,2,3,4)(6,10)(7,14)(8,16)(11,13)(12,18)(15,20);(5,6,7,12,8,10,14,17,9,11,15,18,16,13,20,19);(6,11)(7,15)(10,13)(14,20)Polycyclic presentation:
2^9|a^2=e,b^2=f,d^2=g,f^2=gh,h^2=i,b^a=bd,d^a=dg,f^a=fgh,h^a=hi,c^b=cg,d^b=dh,g^b=gi,h^b=hi,f^c=fi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72221588, #72258306, #72258315, … |
| maximal normal subgroup of | #72258306, #72258315, C4×#11992 |
| maximal quotient of | #72221588 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 34939);