Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #360 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C16 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C8 | #102 |
| commutator | C8 | C22×C16 |
| Frattini, cosocle | C82 | C23 |
| socle | C22 | C2×D4⋊C8 |
| prime radical | C22×D8 | C8 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C8 ◁ C2×C4×C8 ◁ C2×C82 ◁ G | |
| LECS | G ▷ C82 ▷ C42 ▷ C22 ▷ C1 | |
321 conjugacy classes of subgroups; only the 110 normal subgroups are shown.
This group has 321 conjugacy classes of subgroups — too many to draw. Only its 110 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2,3,7,5,10,11,9)(4,13,12,8,15,6,14,16)(17,18,19,22,20,23,25,28,21,24,26,29,27,30,31,32);(2,6)(3,11)(4,14)(7,8)(9,16)(10,13)(12,15);(1,3,5,11)(2,7,10,9)(4,12,15,14)(6,16,13,8)Polycyclic presentation:
2^9|a^2=e,c^2=h,d^2=f,e^2=g,f^2=h,g^2=i,b^a=bd,d^a=df,f^a=fh,c^b=ch,d^b=dfh,e^b=eh,f^b=fh128 conjugacy classes — view the full character table.
| maximal subgroup of | #72287293, #72320389, #72320404, … |
| maximal normal subgroup of | #72287293, #72320389, #72320404, … |
| maximal quotient of | #72287256 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 29662);