Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11663 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#102 |
| commutator | C2×C8 | C2×C42 |
| Frattini, cosocle | C23×C8 | C23 |
| prime radical | C22×#135 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
1046 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2)(3,4)(10,14,15,23)(11,17)(12,20)(16,18,21,24)(19,22);(2,4)(5,6,7,8)(9,10)(11,18)(12,21)(13,15)(14,22)(16,20)(17,23)(19,24);(9,10)(11,18)(12,21)(13,15)(14,22)(16,20)(17,23)(19,24)Polycyclic presentation:
2^9|a^2=f,b^2=g,d^2=h,e^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=eh,h^a=hi,d^b=dhi,e^b=ehi,f^b=fi,h^b=hi,d^c=dhi,e^c=ehi,f^c=fi,h^c=hi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72319728, #72319745, #72319760, … |
| maximal normal subgroup of | #72319728, #72319745, #72319760, … |
| maximal quotient of | #72322472, #72322636 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 53939);