Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #346934 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2846 |
| commutator, Frattini, cosocle | C2×C42 | C24 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
2743 conjugacy classes of subgroups; only the 196 normal subgroups are shown.
This group has 2743 conjugacy classes of subgroups — too many to draw. Only its 196 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) = 64 |
A minimal faithful permutation representation — 4 generators on 16 points:
(2,7)(3,8)(4,5)(10,13)(11,15)(14,16);(1,2,6,3)(4,8,5,7)(11,15)(13,16);(9,10,12,14)(11,16,15,13);(1,3,6,2)(4,8,5,7)Polycyclic presentation:
2^9|b^2=h,c^2=i,d^2=h,e^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,d^a=dg,e^a=eh,f^a=fi,g^a=gh,e^b=eh,f^b=fi,e^c=ei,f^c=fi,g^d=gh59 conjugacy classes — view the full character table.
| Fitting subgroup of | #72272694 |
| maximal subgroup of | #72209905, #72209908, #72226253, … |
| maximal normal subgroup of | #72209905, #72226253, #72272694, … |
| maximal quotient of | #72213214, #72227833, #72228007, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 254142);