Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #19755 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | D42 |
| commutator | C42 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| socle | C22 | D4×C22⋊C4 |
| prime radical | #3475 | C22 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
954 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2,4,7,5,3,8,6)(10,15)(12,13)(16,23)(17,22)(18,24)(19,20);(9,10,14,17)(11,18,21,16)(12,15,13,22)(19,24,20,23);(1,3)(2,5)(4,6)(7,8)(9,11,14,21)(10,16,17,18)(12,19,13,20)(15,23,22,24);(9,12)(11,19)(13,14)(15,22)(20,21)(23,24)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=h,e^2=h,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=eh,f^a=fi,c^b=ch,d^b=de,e^b=eh,f^c=fi,e^d=eh80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 128278);