Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #152686 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #13083 |
| commutator | C2×#41 | C22×C4 |
| Frattini, cosocle | #532 | C23 |
| prime radical | #23994 | C2 |
| LCS | G ▷ C2×#41 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C2×C42 ◁ #3444 ◁ G | |
| LECS | G ▷ #532 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
496 conjugacy classes of subgroups; only the 64 normal subgroups are shown.
This group has 496 conjugacy classes of subgroups — too many to draw. Only its 64 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:
(1,2,3,11,6,12,14,16)(4,10,13,7,5,8,15,9);(2,7,10,11,12,9,8,16)(3,13)(4,5)(14,15);(2,8,12,10)(3,14)(7,16,9,11)(13,15)Polycyclic presentation:
2^9|a^2=f,b^2=dfh,c^2=hi,d^2=i,e^2=i,f^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dg,g^a=gh,h^a=hi,d^b=dg,e^b=ei,f^b=fg,g^b=gh,d^c=di,f^c=fi,f^d=fhi,g^d=gi,g^f=gi41 conjugacy classes — view the full character table.
| maximal subgroup of | #72240970, #72306624, #72306738, … |
| maximal normal subgroup of | #72240970, #72306624, #72306738, … |
| maximal quotient of | #72309236, #72338375, #72339722 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 59885);