Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #483956 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24 |
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 | C2×AGL(1,ℤ/8) |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ C4×#97 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
505 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2)(3,8)(4,6)(5,7)(10,14,16,22)(11,18)(12,20)(15,21,17,24)(19,23);(1,3,5,4)(2,6,7,8);(9,10,11,14,12,17,18,21,13,16,19,22,20,15,23,24);(1,4,5,3)(2,6,7,8)(10,15,16,17)(11,19)(14,21,22,24)(18,23)Polycyclic presentation:
2^9|a^2=g,b^2=e,c^2=f,d^2=eg,f^2=h,h^2=i,b^a=be,c^a=cf,f^a=fh,h^a=hi,d^c=dh,g^c=gi,f^d=fi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72225295, #72225297, #72225298, … |
| maximal normal subgroup of | #72225295, #72225297, #72225298, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 403362);