Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #483919 |
| 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 | #33517 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C22×#97 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
822 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)(3,7)(4,6)(5,8)(9,10,14,19,12,17,23,20)(11,18,21,16,22,15,13,24);(1,3)(2,5)(4,8)(6,7)(9,11,14,21,12,22,23,13)(10,15,19,24,17,18,20,16);(3,8)(5,7)(10,16)(11,21)(13,22)(14,23)(15,20)(17,24)(18,19);(2,6)(5,7)(10,17)(15,18)(16,24)(19,20)Polycyclic presentation:
2^9|a^2=g,b^2=g,f^2=h,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,h^a=hi,c^b=ceh,f^b=fi,f^c=fhi,g^c=gi,h^c=hi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72264870, #72265009, #72265177, … |
| maximal normal subgroup of | #72264870, #72265009, #72265177, … |
| maximal quotient of | #72309872, #72309875, #72310007, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 404032);