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 | |
5989 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 24 points:
(9,10)(11,13)(12,15)(14,16)(19,20)(22,23);(1,2)(3,7)(4,8)(5,6)(10,11)(14,15)(17,18,21,24)(19,22,20,23);(1,3)(2,4)(5,8)(6,7)(9,10,13,11)(12,14,16,15)(17,19,21,20)(18,22,24,23);(2,6)(4,7)(9,11,13,10)(12,15,16,14)(17,18)(19,22)(20,23)(21,24);(1,3)(2,7)(4,6)(5,8)(9,12,13,16)(10,14,11,15)(17,20,21,19)(18,23,24,22);(1,4,5,7)(2,8,6,3)Polycyclic presentation:
2^9|b^2=gh,c^2=h,d^2=g,e^2=h,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^d=fgi,f^e=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8606469);