Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #139424 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #9428 |
| commutator | C4×C8 | C22×C4 |
| Frattini, cosocle | C4×M16 | C23 |
| prime radical | #19745 | C2 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C4×M16 ◁ G | |
| LECS | G ▷ C4×M16 ▷ C2×C4 ▷ C2 ▷ C1 | |
402 conjugacy classes of subgroups; only the 60 normal subgroups are shown.
This group has 402 conjugacy classes of subgroups — too many to draw. Only its 60 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:
(1,2,7,12,16,11,5,10,6,13,15,9,8,3,4,14);(1,3,8,2,6,11,16,13)(4,14,7,9,5,10,15,12);(1,4,6,5)(2,9,13,12)(3,14,11,10)(7,16,15,8)Polycyclic presentation:
2^9|a^2=f,b^2=dfhi,c^2=i,d^2=gh,e^2=gi,f^2=gh,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,g^a=gi,h^a=hi,d^b=dh,e^b=egh,f^b=fhi,g^b=gi,d^c=dgh,e^c=egi,f^c=fgh,h^c=hi,f^d=fi38 conjugacy classes — view the full character table.
| maximal subgroup of | #72231251, #72231258, #72231261, … |
| maximal normal subgroup of | #72231251, #72231258, #72231261, … |
| maximal quotient of | C2×#139427, #72309196, #72338369, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 46623);