Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12028 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#106 |
| commutator | C2×C8 | C2×C42 |
| Frattini, cosocle | C23×C8 | C23 |
| prime radical | C24 | C2×SD16 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
572 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)(5,6,8,7)(10,17,19,18)(11,21,20,22)(12,23)(13,24)(14,15);(2,4)(9,10,14,20,16,19,15,11)(12,21,24,18,13,22,23,17);(5,7,8,6)(9,11,15,19,16,20,14,10)(12,17,23,22,13,18,24,21)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=fh,d^2=h,e^2=h,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=eh,g^a=gi,h^a=hi,d^b=dh,e^b=eh,f^b=fi,h^b=hi,d^c=dh,e^c=eh,f^c=fi,h^c=hi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72319715, #72319728, #72319768, … |
| maximal normal subgroup of | #72319715, #72319728, #72319768, … |
| maximal quotient of | #72322378, #72322431 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 54478);