Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #481837 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#99 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C2×C4⋊C4 | C24 |
| prime radical | #36029 | C2 |
| LCS | G ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×Q8 ◁ C2×#520 ◁ G | |
| LECS | G ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
1105 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)(3,5)(4,6)(7,8)(9,10,14,17,15,20,11,21)(12,18,23,16,13,19,22,24);(2,5)(3,8)(6,7)(9,11,15,14)(10,16,20,24)(12,22,13,23)(17,18,21,19);(2,6)(5,7)(9,11,15,14)(10,17,20,21)(12,23,13,22)(16,18,24,19);(9,12,15,13)(10,18,20,19)(11,22,14,23)(16,21,24,17)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=i,d^2=i,e^2=h,f^2=i,b^a=be,c^a=ch,e^a=eg,g^a=gi,e^b=eh,f^b=fgh,d^c=di,f^e=fi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 389044);