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 | |
6191 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 24 points:
(1,2,6,7)(3,8,5,4)(17,18,22,23)(19,21,20,24);(1,3,6,5)(2,4,7,8)(9,10,14,13)(11,16,12,15)(18,23)(19,20);(9,11,14,12)(10,15,13,16)(17,19,22,20)(18,24,23,21);(2,7)(3,5)(10,13)(15,16)(17,20,22,19)(18,21,23,24);(1,4,6,8)(2,3,7,5)(9,12,14,11)(10,15,13,16)(17,21)(18,19)(20,23)(22,24);(1,5,6,3)(2,4,7,8)(9,13,14,10)(11,16,12,15)Polycyclic presentation:
2^9|a^2=g,b^2=gi,c^2=i,d^2=h,e^2=gi,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^d=fgi,f^e=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8608933);