Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D4×#453 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#120 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #62993 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C4×D4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2969 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 24 points:
(9,10,14,18,15,12,13,17)(11,19,21,20,23,24,22,16);(1,2,4,5)(3,6,7,8)(9,11,15,23)(10,16,12,20)(13,21,14,22)(17,19,18,24);(2,5)(3,7)(9,12)(10,15)(11,20)(13,18)(14,17)(16,23)(19,22)(21,24);(1,3)(2,6)(4,7)(5,8)(9,13,15,14)(10,17,12,18)(11,21,23,22)(16,19,20,24);(2,5)(3,7)Polycyclic presentation:
2^9|a^2=f,b^2=hi,d^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,e^b=eh,f^b=fi,d^c=dh,g^c=gi,e^d=eh80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6503525);