Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24289 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C22×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | D42 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | #2358 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1054 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2)(3,7)(4,5)(6,8)(9,10,13,18,14,19,12,17)(11,16,21,24,22,23,20,15);(2,5)(6,7)(9,11,12,20,14,22,13,21)(10,15,18,16,19,24,17,23);(1,3,4,8)(2,6,5,7)(9,11)(10,16)(12,20)(13,21)(14,22)(15,17)(18,24)(19,23);(2,5)(6,7)(10,17,19,18)(15,23,24,16)Polycyclic presentation:
2^9|a^2=h,b^2=ef,c^2=f,d^2=efh,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,c^b=cg,g^b=gi,g^c=gi92 conjugacy classes — view the full character table.
| maximal quotient of | #72238841, #72238842, #72238843 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 285878);