Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #153123 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2196 |
| commutator, Frattini, cosocle | C2×C4⋊Q8 | C23 |
| prime radical | #31071 | C2 |
| LCS | G ▷ C2×C4⋊Q8 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23 ◁ C22×C4 ◁ C2×C4⋊Q8 ◁ G | |
| LECS | G ▷ C2×C4⋊Q8 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
597 conjugacy classes of subgroups; only the 52 normal subgroups are shown.
This group has 597 conjugacy classes of subgroups — too many to draw. Only its 52 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)(5,6)(7,16)(8,18)(9,19)(10,14)(11,15)(12,17)(13,20);(5,7,11,10)(6,12,15,19)(8,9,20,17)(13,14,18,16);(2,4)(5,6)(7,14)(8,19)(9,18)(10,16)(11,15)(12,20)(13,17)Polycyclic presentation:
2^9|b^2=i,d^2=i,f^2=i,g^2=i,b^a=bd,c^a=ce,d^a=di,f^a=fghi,h^a=hi,c^b=cf,d^b=di,f^b=fi,g^b=gh,d^c=dg,f^c=fi,g^c=gi,h^c=hi,f^d=fhi,g^d=gi,g^f=gi41 conjugacy classes — view the full character table.
| maximal subgroup of | #72228005, #72228026 |
| maximal normal subgroup of | #72228005 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 60320);