Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6360896 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×D42 |
| commutator, Frattini, cosocle | C42 | C25 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
1882 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(2,5)(3,7)(6,8)(10,16)(11,18)(13,14)(17,24)(19,20)(21,22);(2,6)(5,8)(9,10,15,11)(12,20,23,24)(13,18,14,16)(17,21,19,22);(1,2)(3,8)(4,6)(5,7)(9,11,15,10)(12,19,23,17)(13,18,14,16)(20,22,24,21);(9,12)(10,17)(11,19)(13,21)(14,22)(15,23)(16,24)(18,20);(9,13,15,14)(10,18,11,16)(12,21,23,22)(17,20,19,24)Polycyclic presentation:
2^9|b^2=h,c^2=h,e^2=h,f^2=h,g^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fh,g^a=gi,c^b=chi,d^b=df,f^b=fh,e^c=eh,g^c=gi,f^d=fh68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6268102);