Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2104 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #401 |
| commutator | C24 | C22×C8 |
| Frattini, cosocle | C24×C4 | C23 |
| socle | C22 | #1443 |
| prime radical | C25 | C22×C4 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C22 ▷ C1 | |
663 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2,3,5,4,6,7,8)(9,10)(11,14,18,22)(12,19)(13,16)(15,20)(17,21,24,23);(9,11)(10,14,16,22)(12,17,20,24)(13,18)(15,23)(19,21);(9,12)(10,15,16,19)(11,17,18,24)(13,20)(14,21)(22,23)Polycyclic presentation:
2^9|a^2=g,b^2=de,c^2=d,g^2=i,b^a=bd,c^a=ce,d^a=dh,f^a=fh,c^b=cf,d^b=dh,e^b=eh,f^b=fh,g^b=gh,e^c=eh,f^c=fh92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 28281);