Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11836 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1514 |
| commutator | C2×C8 | C2×C42 |
| Frattini, cosocle | C22×M16 | C23 |
| prime radical | C24 | AGL(1,ℤ/8) |
| LCS | G ▷ C2×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C4×M16 ◁ G | |
| LECS | G ▷ C22×M16 ▷ C2×C4 ▷ C2 ▷ C1 | |
415 conjugacy classes of subgroups; only the 130 normal subgroups are shown.
This group has 415 conjugacy classes of subgroups — too many to draw. Only its 130 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)(8,17)(11,18)(12,15)(13,19)(16,20);(5,6)(7,15,16,18)(8,11)(9,13)(10,14,19,20)(12,17);(6,11,13,12)(7,16)(10,18,19,15)(14,20)Polycyclic presentation:
2^9|a^2=e,b^2=f,c^2=ghi,d^2=gi,g^2=i,h^2=i,b^a=bd,d^a=dg,f^a=fghi,g^a=gi,h^a=hi,c^b=cg,d^b=dh,g^b=gi,d^c=di,f^d=fi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72240808, #72250331, #72250372, … |
| maximal normal subgroup of | #72240808, #72250331, #72250372, … |
| maximal quotient of | #72240811, #72264539 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 34782);