Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #149899 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×AΣL(1,𝔽16) |
| commutator | C22×Q8 | C22×C4 |
| Frattini, cosocle | C2×C4⋊Q8 | C23 |
| prime radical | #3486 | C4 |
| LCS | G ▷ C22×Q8 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C23×C4 ◁ C22×C4⋊Q8 ◁ G | |
| LECS | G ▷ C2×C4⋊Q8 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
642 conjugacy classes of subgroups; only the 72 normal subgroups are shown.
This group has 642 conjugacy classes of subgroups — too many to draw. Only its 72 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:
(1,2)(3,4)(5,6,7,12,9,14,16,20)(8,13,18,11,17,19,15,10);(6,10,19,12,14,11,13,20)(7,15)(8,17)(16,18);(2,4)Polycyclic presentation:
2^9|a^2=f,b^2=dfh,d^2=i,f^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dg,g^a=gh,h^a=hi,d^b=dg,f^b=fg,g^b=gh,f^d=fhi,g^d=gi,g^f=gi53 conjugacy classes — view the full character table.
| maximal subgroup of | #72279392, #72283498, #72283624 |
| maximal normal subgroup of | #72279392, #72283498, #72283624 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 59850);