Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24289 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C4 |
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 | C24 | H(5,𝔽2) |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
921 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)(4,6)(5,8)(9,10,14,19,15,20,13,18)(11,21,23,24,12,17,22,16);(1,3,5,4)(2,6,8,7)(9,11,13,22,15,12,14,23)(10,16,19,21,20,24,18,17);(1,4,5,3)(2,7,8,6)(9,12)(10,17)(11,15)(13,23)(14,22)(16,19)(18,24)(20,21);(2,8)(6,7)(10,18,20,19)(16,17,24,21)Polycyclic presentation:
2^9|a^2=h,b^2=e,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 | #72238842, #72238843, #72238844 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 285688);