Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #133259 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #8522 |
| commutator | C2×C42 | C22×C4 |
| Frattini, cosocle | C4×M16 | C23 |
| prime radical | #60798 | C2 |
| LCS | G ▷ C2×C42 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C4×M16 ◁ G | |
| LECS | G ▷ C4×M16 ▷ C2×C4 ▷ C2 ▷ C1 | |
783 conjugacy classes of subgroups; only the 76 normal subgroups are shown.
This group has 783 conjugacy classes of subgroups — too many to draw. Only its 76 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:
(1,2,3,9,4,6,12,14,5,7,11,15,13,8,10,16);(2,6)(3,10)(4,13)(7,8)(9,15)(11,12);(2,7)(6,8)Polycyclic presentation:
2^9|a^2=f,d^2=h,f^2=g,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,h^a=hi,d^b=dh,f^b=fghi,g^b=gi,f^c=fh,f^d=fi,f^e=fi38 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#133259, #72240968, #72240971, … |
| maximal normal subgroup of | C2×#133259, #72240968, #72240971, … |
| maximal quotient of | #72219581, C2×#133259, #72279219, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 40455);