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 | |
5998 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 24 points:
(1,2,6,5)(3,8,4,7)(10,15)(11,12)(17,18)(19,23)(20,21)(22,24);(1,3,6,4)(2,7,5,8)(9,10)(11,16)(12,13)(14,15)(17,19)(18,23)(20,24)(21,22);(9,11,14,12)(10,16,15,13)(17,20)(18,19)(21,24)(22,23);(2,5)(7,8)(9,12,14,11)(10,16,15,13)(17,21,22,19)(18,20,24,23);(1,4,6,3)(2,8,5,7);(1,5,6,2)(3,8,4,7)(9,13,14,16)(10,12,15,11)(17,20)(18,21)(19,24)(22,23)Polycyclic presentation:
2^9|a^2=ghi,b^2=ghi,c^2=hi,d^2=h,e^2=ghi,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,f^d=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8364216);