Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3290 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | D42 |
| commutator | C2×C4 | C22×C42 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×H(5,𝔽2) | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2389 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(2,4)(5,6)(7,8)(9,10,15,19,16,21,14,20)(11,22,23,13,18,24,17,12);(1,2)(3,4)(9,11,15,23,16,18,14,17)(10,12,20,24,21,13,19,22);(6,8)(9,12)(10,17)(11,20)(13,16)(14,22)(15,24)(18,19)(21,23);(2,4)(9,13)(10,18)(11,21)(12,16)(14,22)(15,24)(17,19)(20,23)Polycyclic presentation:
2^9|a^2=g,b^2=h,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,e^a=ei,h^a=hi,d^b=dei,f^c=fi,g^c=gi,e^d=ei,h^d=hi116 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 263268);