Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C26 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
5397 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 24 points:
(1,2)(3,7)(4,5)(6,8)(18,21)(19,20);(1,3,6,5)(2,7,8,4)(9,10,13,14)(11,15,16,12)(17,18,22,21)(19,24,20,23);(2,8)(3,5)(10,14)(12,15)(17,19,22,20)(18,23,21,24);(9,11)(10,12)(13,16)(14,15)(17,20,22,19)(18,24,21,23);(1,4)(2,3)(5,8)(6,7)(17,21,22,18)(19,24,20,23);(1,5,6,3)(2,4,8,7)(9,12)(10,11)(13,15)(14,16)Polycyclic presentation:
2^9|b^2=i,c^2=g,d^2=g,e^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^c=fgi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8089244);