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 | |
5341 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 24 points:
(9,10,14,12)(11,16,13,15)(17,18,22,20)(19,23,21,24);(1,2)(3,7)(4,5)(6,8)(9,11,14,13)(10,15,12,16)(17,19,22,21)(18,23,20,24);(2,5)(6,7)(9,10,14,12)(11,15,13,16)(17,19,22,21)(18,24,20,23);(3,8)(6,7)(9,11,14,13)(10,15,12,16)(17,20,22,18)(19,23,21,24);(1,3)(2,6)(4,8)(5,7)(9,12,14,10)(11,15,13,16)(17,21,22,19)(18,23,20,24);(1,2)(3,6)(4,5)(7,8)(9,13,14,11)(10,15,12,16)Polycyclic presentation:
2^9|a^2=gh,b^2=gh,c^2=gh,d^2=gh,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,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, 8255160);