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 | |
5292 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 24 points:
(2,4)(6,8)(9,10,13,12)(11,15,16,14)(17,18,22,20)(19,24,23,21);(3,7)(6,8)(10,12)(14,15)(18,20)(19,23);(1,2,5,4)(3,6,7,8)(17,19)(18,21)(20,24)(22,23);(1,3)(2,6)(4,8)(5,7)(9,11)(10,14)(12,15)(13,16)(17,19)(18,21)(20,24)(22,23);(1,3)(2,6)(4,8)(5,7)(9,12,13,10)(11,15,16,14)(17,20,22,18)(19,24,23,21);(1,4,5,2)(3,8,7,6)(9,11)(10,14)(12,15)(13,16)(17,21,22,24)(18,23,20,19)Polycyclic presentation:
2^9|a^2=g,c^2=hi,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fghi,c^b=ci,d^b=dh,e^b=eghi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8000452);