Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #453 |
| automorphism | ? |
| homology | Abelianization (H1) = C43 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | #120 |
| commutator | C2×C4 | C43 |
| Frattini, cosocle | C24×C4 | C23 |
| socle | C23 | #386 |
| prime radical | C25 | C2×D4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C2 ▷ C1 | |
1950 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2,3,4)(9,10)(11,18)(12,21)(13,16)(14,17)(15,19)(20,24)(22,23);(9,11,13,19,14,20,12,22)(10,15,16,24,17,23,21,18);(5,6,7,8)(9,10)(11,15)(12,21)(13,16)(14,17)(18,22)(19,24)(20,23)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=h,d^2=i,g^2=i,b^a=bd,c^a=ce,d^a=di,d^b=di,e^b=ei,d^c=di128 conjugacy classes — view the full character table.
| maximal subgroup of | #72235145, #72235150, C2×#123488, … |
| maximal normal subgroup of | #72235145, #72235150, C2×#123488, … |
| maximal quotient of | C2×#123488, #72235278, #72238849 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 30684);