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:
(1,2,5,6)(3,7,8,4)(9,10)(11,12)(13,16)(14,15)(17,18)(19,21)(20,23)(22,24);(1,2)(3,4)(5,6)(7,8)(9,11,13,15)(10,14,16,12)(17,18,22,24)(19,21,23,20);(3,8)(4,7)(9,10,13,16)(11,14,15,12)(17,19,22,23)(18,20,24,21);(2,6)(3,8)(9,10,13,16)(11,12,15,14)(17,20,22,21)(18,19,24,23);(1,3)(2,7)(4,6)(5,8)(9,10)(11,14)(12,15)(13,16)(17,18)(19,20)(21,23)(22,24);(1,4,5,7)(2,3,6,8)(9,12,13,14)(10,15,16,11)(17,21,22,20)(18,23,24,19)Polycyclic presentation:
2^9|a^2=gh,b^2=h,c^2=h,d^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,e^c=egi,f^c=fgi,e^d=egi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8947805);