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 | |
5555 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)(3,4)(5,6)(7,8)(9,10)(11,15)(12,13)(14,16)(17,18,22,20)(19,24,21,23);(1,3,7,6)(2,4,8,5)(10,12)(14,15);(1,4,7,5)(2,6,8,3)(17,19,22,21)(18,23,20,24);(1,4,7,5)(2,3,8,6)(9,11,13,16)(10,14,12,15)(17,19,22,21)(18,24,20,23);(1,5,7,4)(2,3,8,6)(9,12,13,10)(11,14,16,15)(17,20,22,18)(19,24,21,23);(1,6,7,3)(2,4,8,5)(9,11)(10,15)(12,14)(13,16)(17,21,22,19)(18,24,20,23)Polycyclic presentation:
2^9|a^2=hi,b^2=i,c^2=h,d^2=gh,e^2=gh,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=egi,e^d=eghi,f^d=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10329941);