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 | |
2214 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(2,4)(5,6,7,8)(9,10,14,20)(11,17,12,22)(13,18,15,19)(16,23,24,21);(9,11,13,21,14,12,15,23)(10,16,19,17,20,24,18,22);(1,2,3,4)(9,12,14,11)(10,17,20,22)(13,23,15,21)(16,19,24,18);(9,13,14,15)(10,18,20,19)(11,21,12,23)(16,22,24,17)Polycyclic presentation:
2^9|a^2=h,b^2=e,c^2=f,d^2=i,e^2=i,g^2=i,b^a=be,c^a=cf,e^a=ei,c^b=cg,g^b=gi,g^c=gi104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 265681);