Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #1357 |
| automorphism | ? |
| homology | Abelianization (H1) = C4×C8 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #331 |
| commutator | C42 | C4×C8 |
| Frattini, cosocle | C2×C43 | C22 |
| socle | C22 | #1270 |
| prime radical | C24 | #99 |
| LCS | G ▷ C42 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ C2×C43 ◁ G | |
| LECS | G ▷ C2×C43 ▷ C22×C4 ▷ C2 ▷ C1 | |
454 conjugacy classes of subgroups; only the 122 normal subgroups are shown.
This group has 454 conjugacy classes of subgroups — too many to draw. Only its 122 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 — 2 generators on 20 points:
(1,2,3,4)(5,6)(7,14)(8,11)(9,13)(10,15)(12,18)(16,19)(17,20);(5,7)(6,10,11,19,13,20,12,14)(8,15)(9,16)(17,18)Polycyclic presentation:
2^9|a^2=d,b^2=e,c^2=f,e^2=h,g^2=i,b^a=bc,c^a=cf,e^a=efg,g^a=gi,h^a=hi,c^b=cg,f^b=fi,g^b=gi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72227993 |
| maximal normal subgroup of | #72227993 |
| maximal quotient of | #72237311, #72237312, #72237313, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 576);