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 | |
5443 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 24 points:
(1,2,6,4)(3,7,5,8)(9,10)(11,14)(12,15)(13,16)(17,18,21,20)(19,23,24,22);(2,4)(7,8)(9,11,13,12)(10,14,16,15)(17,19,21,24)(18,22,20,23);(1,2,6,4)(3,7,5,8)(9,10,13,16)(11,14,12,15)(17,18,21,20)(19,22,24,23);(1,3,6,5)(2,7,4,8)(9,10)(11,14)(12,15)(13,16);(1,4,6,2)(3,7,5,8);(1,5,6,3)(2,7,4,8)(9,12,13,11)(10,15,16,14)(17,20,21,18)(19,23,24,22)Polycyclic presentation:
2^9|a^2=g,b^2=ghi,c^2=gh,d^2=i,e^2=i,f^2=gh,b^a=bg,c^a=ch,f^a=fi,c^b=ci,e^b=ei,d^c=dh,f^c=fi,e^d=ei110 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7591301);