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 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1847 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2)(3,5)(4,6)(7,8)(9,10,14,18,15,19,13,23)(11,20,21,12,17,24,22,16);(1,2)(3,7)(4,6)(5,8)(9,11,13,22,15,17,14,21)(10,16,18,20,19,12,23,24);(1,3)(2,5)(4,8)(6,7)(9,10)(11,12)(13,18)(14,23)(15,19)(16,17)(20,21)(22,24);(3,8)(5,7)(9,12)(10,17)(11,19)(13,20)(14,24)(15,16)(18,22)(21,23);(2,6)(5,7)(9,13,15,14)(10,18,19,23)(11,21,17,22)(12,20,16,24)Polycyclic presentation:
2^9|a^2=g,b^2=f,e^2=i,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,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, 6910257);