Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #1357 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C8 | #331 |
| commutator | C42 | C22×C8 |
| Frattini, cosocle | C43 | C23 |
| socle | C2 | #7569 |
| prime radical | #3147 | C4 |
| LCS | G ▷ C42 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8 ◁ C22×C8 ◁ C42×C8 ◁ G | |
| LECS | G ▷ C43 ▷ C22×C4 ▷ C2 ▷ C1 | |
384 conjugacy classes of subgroups; only the 110 normal subgroups are shown.
This group has 384 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) = 32 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2)(3,12,14,22,15,21,18,7)(4,8)(5,9)(6,11)(10,19)(13,26,27,32,28,31,29,20)(16,23)(17,25)(24,30);(1,3)(2,7)(4,13)(5,18)(6,15)(8,20)(9,21)(10,12)(11,22)(14,19)(16,29)(17,28)(23,31)(24,26)(25,32)(27,30);(1,4,5,16,6,17,19,30)(2,8,9,23,11,25,10,24)(3,13,18,29,15,28,14,27)(7,20,21,31,22,32,12,26)Polycyclic presentation:
2^9|a^2=e,c^2=fgh,d^2=g,e^2=h,f^2=i,b^a=bd,d^a=df,f^a=fi,g^a=gi,d^b=dg,e^b=efgi,f^b=fi,h^b=hi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72225426, #72225431, #72225442, … |
| maximal normal subgroup of | #72225426, #72225431, #72225442, … |
| maximal quotient of | #72227208 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 31409);