Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #13681 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2196 |
| commutator | C4⋊Q8 | C22×C4 |
| Frattini, cosocle | C2×C4⋊Q8 | C23 |
| prime radical | C2×#127 | D4 |
| LCS | G ▷ 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 | |
524 conjugacy classes of subgroups; only the 62 normal subgroups are shown.
This group has 524 conjugacy classes of subgroups — too many to draw. Only its 62 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:
(2,4)(5,6,10,15,11,13,8,14)(7,16,19,12,18,20,17,9);(1,2,3,4)(5,7)(6,12)(8,19)(9,13)(10,17)(11,18)(14,20)(15,16);(6,13)(8,16)(9,12)(10,20)(14,17)(15,19)Polycyclic presentation:
2^9|a^2=f,b^2=df,d^2=i,e^2=h,f^2=i,g^2=i,b^a=bd,c^a=ce,e^a=eghi,g^a=gh,h^a=hi,d^b=di,e^b=eghi,f^b=fi,h^b=hi,d^c=dg,e^c=eh,f^c=fg,g^c=gi,e^d=eh,g^d=gi,f^e=fh,g^f=gi50 conjugacy classes — view the full character table.
| maximal subgroup of | #72283566, #72283624, #72283688, … |
| maximal normal subgroup of | #72283566, #72283624, #72283688, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 59782);