Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417530 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2403 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#535 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1463 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2,5,4)(3,7,8,6)(9,10)(11,17,19,14)(12,16)(13,15)(18,24,21,22)(20,23);(1,3)(2,6)(4,7)(5,8)(11,18)(12,20)(14,22)(16,23)(17,24)(19,21);(1,4,5,2)(3,7,8,6)(9,11)(10,14)(12,21)(13,19)(15,17)(16,24)(18,20)(22,23);(3,8)(6,7)(9,10)(11,14)(12,16)(13,15)(17,19)(18,22)(20,23)(21,24)Polycyclic presentation:
2^9|a^2=eh,c^2=e,g^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,c^b=cg,d^b=de,g^b=gi,g^c=gi,h^c=hi,f^d=fi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72252550, #72252557, #72252575, … |
| maximal normal subgroup of | #72252550, #72252557, #72252575, … |
| maximal quotient of | C2×#449630 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 356826);