Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12525 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #493 |
| commutator | C4×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C22 | #1957 |
| prime radical | C23 | #486 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ C2×C82 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
320 conjugacy classes of subgroups; only the 118 normal subgroups are shown.
This group has 320 conjugacy classes of subgroups — too many to draw. Only its 118 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) = 256 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2,4,7,6,10,5,9)(3,11,13,16,12,15,14,8)(18,22)(19,26)(20,27)(23,28)(24,32)(25,31)(29,30);(2,7,10,9)(3,12)(8,15,16,11)(13,14)(17,18,23,32,20,24,19,22,21,25,26,30,27,29,28,31);(1,3,5,14,6,12,4,13)(2,8,7,11,10,16,9,15)Polycyclic presentation:
2^9|a^2=f,b^2=dfgi,c^2=e,d^2=gi,e^2=h,f^2=h,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,g^a=gi,c^b=ch92 conjugacy classes — view the full character table.
| maximal subgroup of | #72210421 |
| maximal normal subgroup of | #72210421 |
| maximal quotient of | #72210378 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 49027);