Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #149794 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2126 |
| commutator | C2×C42 | C22×C4 |
| Frattini, cosocle | #423 | C23 |
| socle | C2 | #11507 |
| prime radical | #3444 | C4 |
| LCS | G ▷ C2×C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C42 ◁ C2×C42 ◁ #2920 ◁ G | |
| LECS | G ▷ #423 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
417 conjugacy classes of subgroups; only the 56 normal subgroups are shown.
This group has 417 conjugacy classes of subgroups — too many to draw. Only its 56 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 16 |
A minimal faithful permutation representation — 3 generators on 16 points:
(1,2,5,10,6,11,3,8,7,12,14,15,4,9,13,16);(1,3)(2,8)(4,5)(6,14)(7,13)(9,10)(11,15)(12,16);(2,9,12,11)(3,13)(5,14)(8,15,16,10)Polycyclic presentation:
2^9|a^2=e,c^2=h,d^2=hi,e^2=g,f^2=i,g^2=i,b^a=bd,c^a=cg,d^a=df,f^a=fh,h^a=hi,c^b=ci,d^b=dhi,e^b=efh,f^b=fi,e^c=ei,e^d=ehi,f^e=fi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72272670, #72272672, #72272673 |
| maximal normal subgroup of | #72272670, #72272672, #72272673 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 56990);