Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #8626 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #380 |
| commutator | C22×C4 | C2×C42 |
| Frattini, cosocle | C24×C4 | C23 |
| socle | C22 | C22⋊#95 |
| prime radical | C2×#525 | C22 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C22 ▷ C1 | |
1310 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2)(3,7,8,5)(4,6)(9,10)(11,18)(12,21)(13,17)(14,19)(15,20)(16,23)(22,24);(1,3)(2,5)(4,8)(6,7)(9,11)(10,14,17,18)(12,22)(13,19)(15,16,24,21)(20,23);(1,3)(2,5)(4,8)(6,7)(9,11)(10,15)(12,22)(13,19)(14,16)(17,24)(18,21)(20,23)Polycyclic presentation:
2^9|a^2=f,b^2=g,e^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ehi,g^a=gi,d^b=di,e^b=ei,f^b=fh,d^c=di,e^c=ei,f^c=fh80 conjugacy classes — view the full character table.
| maximal subgroup of | #72260815, #72260823, #72260837, … |
| maximal normal subgroup of | #72260815, #72260823, #72260837, … |
| maximal quotient of | #72261426 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 13249);