Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6993875 |
| 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 | |
1780 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)(9,10,15,17,14,11,13,18)(12,16,21,20,23,19,22,24);(1,2,5,6)(3,4,8,7)(9,11,14,10)(12,20,23,24)(13,17,15,18)(16,21,19,22);(1,3)(2,7)(4,6)(5,8)(9,10,14,11)(12,16,23,19)(13,17,15,18)(20,21,24,22);(2,6)(3,8)(9,12)(10,16)(11,19)(13,22)(14,23)(15,21)(17,20)(18,24);(1,4,5,7)(2,3,6,8)Polycyclic presentation:
2^9|a^2=gi,b^2=hi,c^2=i,e^2=h,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh,g^c=gi,f^d=fi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7089190);