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 | |
5998 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 24 points:
(2,5)(6,8)(10,14)(12,16)(17,18,23,21)(19,20,24,22);(3,7)(6,8)(10,14)(11,15)(18,21)(19,24);(1,2)(3,6)(4,5)(7,8)(9,10)(11,12)(13,14)(15,16)(17,19)(18,22)(20,21)(23,24);(10,14)(11,15)(17,20,23,22)(18,19,21,24);(1,3)(2,6)(4,7)(5,8)(9,11,13,15)(10,12,14,16)(17,21,23,18)(19,20,24,22);(1,2)(3,6)(4,5)(7,8)(9,12)(10,15)(11,14)(13,16)(17,22,23,20)(18,24,21,19)Polycyclic presentation:
2^9|a^2=g,d^2=g,e^2=i,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,f^c=fgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8819516);