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 | |
5184 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 24 points:
(2,6)(4,7)(10,14)(15,16)(18,21)(22,23);(1,2,5,6)(3,4,8,7)(9,10,13,14)(11,15,12,16);(2,6)(3,8)(17,18)(19,23)(20,21)(22,24);(9,11,13,12)(10,15,14,16)(17,19,20,24)(18,22,21,23);(1,3,5,8)(2,7,6,4);(1,4,5,7)(2,8,6,3)(9,12,13,11)(10,16,14,15)Polycyclic presentation:
2^9|b^2=g,d^2=ghi,e^2=i,f^2=g,b^a=bg,c^a=ch,f^a=fi,c^b=ci,e^b=ei,d^c=dh,e^c=ei110 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7585921);