Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6502819 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#121 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2067 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 24 points:
(9,10,13,17,14,18,12,16)(11,19,23,15,22,24,21,20);(1,2)(3,7)(4,6)(5,8)(9,11,13,23,14,22,12,21)(10,15,16,19,18,20,17,24);(1,3)(2,5)(4,8)(6,7)(9,10)(11,20)(12,17)(13,16)(14,18)(15,22)(19,21)(23,24);(3,8)(5,7)(9,12,14,13)(10,16,18,17)(11,21,22,23)(15,19,20,24);(2,6)(5,7)(11,22)(15,20)(19,24)(21,23)Polycyclic presentation:
2^9|a^2=g,b^2=f,d^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,e^b=eh,d^c=dh,g^c=gi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6410019);