Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30412 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22×D4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C23 | C22×C22⋊C4 |
| prime radical | #34738 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C2×C4×C22⋊C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2176 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)(9,10,14,16,13,17,12,21)(11,18,20,15,19,24,22,23);(1,3,4,2)(5,6)(7,8)(11,19)(15,23)(18,24)(20,22);(6,8)(9,11)(10,15)(12,22)(13,19)(14,20)(16,24)(17,23)(18,21);(2,3)(9,11)(10,15)(12,22)(13,19)(14,20)(16,24)(17,23)(18,21)Polycyclic presentation:
2^9|a^2=h,b^2=e,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=def,f^a=fi,c^b=cg,d^b=dei,f^c=fi,f^d=fi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72260037, #72260045, #72262491 |
| maximal normal subgroup of | #72260037, #72260045, #72262491 |
| maximal quotient of | #72258596 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 290453);