Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #394335 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#121 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×AGL(1,ℤ/8) | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1301 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)(3,4)(6,7)(9,10,15,21,13,12,14,17)(11,18,22,23,19,16,20,24);(5,6,8,7)(9,11,14,20,13,19,15,22)(10,16,17,23,12,18,21,24);(5,6,8,7)(9,12,13,10)(11,18,19,16)(14,17,15,21)(20,23,22,24);(2,4)(5,7,8,6)Polycyclic presentation:
2^9|a^2=ef,b^2=f,c^2=e,d^2=ei,f^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,g^b=gi,f^c=fi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 304897);