Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6996395 |
| 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 | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1786 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 24 points:
(2,6)(4,7)(9,10,15,18)(11,19,22,13)(12,23,17,14)(16,21,24,20);(1,2,5,6)(3,4,8,7)(9,11)(10,16)(12,19)(13,17)(14,21)(15,22)(18,24)(20,23);(1,3)(2,7)(4,6)(5,8)(9,12,14,10,15,17,23,18)(11,19,21,16,22,13,20,24);(2,6)(3,8)(9,13)(10,11)(12,20)(14,24)(15,19)(16,23)(17,21)(18,22);(1,4,5,7)(2,3,6,8)Polycyclic presentation:
2^9|a^2=i,b^2=h,c^2=g,e^2=h,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,g^a=gi,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh,f^d=fi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7092622);