Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #198162 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#386 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#121 | C22 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
2154 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)(3,5,7,8)(4,6)(9,10,14,18,15,11,13,17)(12,20,22,16,23,24,21,19);(1,3,4,7)(2,5)(6,8)(9,11,15,10)(12,19,23,16)(13,17,14,18)(20,22,24,21);(9,12,15,23)(10,16,11,19)(13,22,14,21)(17,24,18,20);(2,6)(3,7)(10,11)(16,19)(17,18)(20,24)Polycyclic presentation:
2^9|a^2=g,b^2=egi,c^2=hi,e^2=hi,f^2=hi,g^2=hi,b^a=be,c^a=cf,d^a=di,e^a=eh,d^b=di,e^b=ei,f^b=fhi,g^b=gi,e^c=ehi,f^c=fhi,g^c=ghi74 conjugacy classes — view the full character table.
| maximal subgroup of | #72257736, #72257744, #72257754, … |
| maximal normal subgroup of | #72257736, #72257744, #72257754, … |
| maximal quotient of | #72264106, #72264321 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 105666);