Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 6 |
| isoclinism | isoclinic class of #94630 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #7394 |
| commutator | #407 | C2×C4 |
| Frattini, cosocle | #1783 | C22 |
| prime radical | #10326 | C2 |
| LCS | G ▷ #407 ▷ C23×C4 ▷ C23 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22 ◁ C22×C4 ◁ C23×C4 ◁ #1783 ◁ G | |
| LECS | G ▷ #1783 ▷ C23×C4 ▷ C23 ▷ C22 ▷ C2 ▷ C1 | |
360 conjugacy classes of subgroups; only the 22 normal subgroups are shown.
This group has 360 conjugacy classes of subgroups — too many to draw. Only its 22 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 — 2 generators on 16 points:
(1,2)(3,6)(4,14,11,8)(5,9)(7,12,15,10)(13,16);(2,6)(3,10)(4,11)(7,8)(9,16)(12,13)(14,15)Polycyclic presentation:
2^9|a^2=d,c^2=fi,e^2=i,f^2=i,b^a=bc,c^a=ce,e^a=eg,f^a=fh,g^a=gi,h^a=hi,c^b=cf,d^b=defi,e^b=ei,f^b=fi,g^b=gh,d^c=dgi,e^c=ehi,g^c=gi,e^d=ei,f^d=fi29 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1826);