Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #484095 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#442 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| prime radical | C23×D8 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ #1752 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
756 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2)(3,5)(4,6)(7,8)(10,14,16,23)(11,18)(12,21)(15,22,24,17)(19,20);(1,3,4,7)(2,5,6,8)(9,10)(11,15)(12,22)(13,16)(14,19)(17,21)(18,23)(20,24);(2,6)(5,8);(10,15)(11,19)(12,21)(14,22)(16,24)(17,23)(18,20)Polycyclic presentation:
2^9|a^2=g,b^2=f,e^2=h,h^2=i,b^a=be,c^a=cf,e^a=eh,h^a=hi,d^b=de,e^b=ehi,g^b=gi,h^b=hi,e^d=ehi,h^d=hi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72264898, #72264905, #72264915, … |
| maximal normal subgroup of | #72264898, #72264905, #72264915, … |
| maximal quotient of | #72309672, #72309709, #72309736, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 404897);