Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2002 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | #120 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C23 | C2×#120 |
| prime radical | C25 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1866 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(5,6,7,8)(9,10,13,16,14,18,12,17)(11,15,19,22,20,24,21,23);(1,2)(3,4)(9,11,13,21,14,20,12,19)(10,15,17,22,18,24,16,23);(2,4)(10,16,18,17)(11,19,20,21);(9,12,14,13)(10,17,18,16)(11,19,20,21)(15,22,24,23)Polycyclic presentation:
2^9|a^2=h,b^2=f,c^2=e,d^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,e^a=ei,f^a=fi,c^b=cg,e^b=ei,g^b=gi,h^b=hi104 conjugacy classes — view the full character table.
| maximal subgroup of | #72239940 |
| maximal normal subgroup of | #72239940 |
| maximal quotient of | #72239944 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 265483);