Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #188035 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#379 |
| commutator | C24 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×D42 | C2 |
| LCS | G ▷ C24 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
5418 conjugacy classes of subgroups; only the 358 normal subgroups are shown.
This group has 5418 conjugacy classes of subgroups — too many to draw. Only its 358 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) = 32 |
A minimal faithful permutation representation — 4 generators on 16 points:
(1,2,4,8,5,6,3,7)(9,10,11,14,12,13,15,16);(2,6)(3,4)(10,13)(11,15);(10,13)(11,15);(1,3,5,4)(2,7,6,8)Polycyclic presentation:
2^9|a^2=g,d^2=hi,g^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fi,d^b=dhi,g^b=gh,g^c=gi74 conjugacy classes — view the full character table.
| maximal subgroup of | #72226358, #72232781, #72232782, … |
| maximal normal subgroup of | #72226358, #72232781, #72232782, … |
| maximal quotient of | #72225881 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 95347);