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 | |
5179 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 24 points:
(1,2,5,3)(4,6,8,7)(9,10)(11,15)(12,13)(14,16)(17,18,20,22)(19,21,23,24);(2,3)(4,8)(9,11)(10,15)(12,16)(13,14)(17,18)(19,21)(20,22)(23,24);(1,2,5,3)(4,7,8,6)(10,16)(11,13)(17,19,20,23)(18,21,22,24);(1,2,5,3)(4,6,8,7)(9,11,14,13)(10,15,16,12);(1,3,5,2)(4,7,8,6)(9,12)(10,11)(13,16)(14,15);(1,4)(2,6)(3,7)(5,8)(9,13,14,11)(10,12,16,15)Polycyclic presentation:
2^9|a^2=g,c^2=g,d^2=i,e^2=hi,f^2=h,b^a=bg,c^a=ch,e^a=eh,c^b=ci,d^b=di,e^b=ei,f^b=fi,d^c=dh,f^c=fi104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7713591);