Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #125644 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1595 |
| commutator, Frattini, cosocle | C2×#97 | C23 |
| LCS | G ▷ C2×#97 ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×#97 ◁ G | |
| LECS | G ▷ C2×#97 ▷ C22×C4 ▷ C2 ▷ C1 | |
587 conjugacy classes of subgroups; only the 56 normal subgroups are shown.
This group has 587 conjugacy classes of subgroups — too many to draw. Only its 56 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2)(3,5)(4,7)(6,8)(9,10)(11,14)(12,21)(13,17)(15,18)(16,20)(19,22)(23,24);(10,14)(11,18)(12,20)(15,16)(17,22)(19,24)(21,23);(2,5)(3,8)(6,7)(11,19)(18,24)Polycyclic presentation:
2^9|d^2=fi,e^2=h,f^2=i,b^a=bd,c^a=ce,d^a=df,e^a=eh,f^a=fi,g^a=gi,d^b=df,e^b=egi,f^b=fi,d^c=dg,e^c=eh,e^d=ei41 conjugacy classes — view the full character table.
| maximal subgroup of | #72213794, #72265135, #72265142, … |
| maximal normal subgroup of | #72213794, #72265135, #72265142, … |
| maximal quotient of | #72213167, #72213171 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 32840);