Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #19671 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#127 |
| commutator | C42 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| socle | C22 | C2×#390 |
| prime radical | C2×#3147 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1277 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2)(3,4)(5,8)(6,7)(9,10,15,18,13,17,14,23)(11,19,21,24,20,16,12,22);(1,3,4,2,6,8,5,7)(9,11,14,12,13,20,15,21)(10,16,23,24,17,19,18,22);(9,12,14,20,13,21,15,11)(10,16,18,22,17,19,23,24);(1,4,6,5)(2,3,7,8)Polycyclic presentation:
2^9|a^2=g,b^2=egh,c^2=f,d^2=h,e^2=h,f^2=i,g^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fi,c^b=ci,d^b=dh80 conjugacy classes — view the full character table.
| maximal subgroup of | #72310172, #72310174, #72310177, … |
| maximal normal subgroup of | #72310172, #72310174, #72310177, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 113655);