Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #484095 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#442 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C4×C8 | C24 |
| prime radical | C22×D16 | C4 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ #1451 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
491 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2,4,7,5,3,6,8)(10,14,16,23)(11,18)(12,19)(15,21,24,17)(20,22);(9,10)(11,15)(12,21)(13,16)(14,22)(17,19)(18,23)(20,24);(1,3,4,8,5,2,6,7)(9,11,19,22,13,20,12,18)(10,15,17,14,16,24,21,23);(2,3)(7,8)Polycyclic presentation:
2^9|a^2=f,c^2=fgi,e^2=g,f^2=h,g^2=i,b^a=be,c^a=cg,d^a=dh,e^a=eg,g^a=gi,e^b=egi,f^b=fi,g^b=gi,d^c=dh,e^c=ei62 conjugacy classes — view the full character table.
| maximal subgroup of | #72216715, #72216752, #72216759, … |
| maximal normal subgroup of | #72216715, #72216752, #72216759, … |
| maximal quotient of | #72225063, #72225100, #72227128, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 391293);