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 | |
5117 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 24 points:
(2,3)(6,7)(9,10)(11,13)(12,14)(15,16)(17,18,21,20)(19,23,24,22);(1,2,5,3)(4,6,8,7)(17,19,21,24)(18,22,20,23);(2,3)(6,7)(9,10,12,14)(11,13,15,16);(2,3)(4,8)(11,15)(13,16);(1,3,5,2)(4,7,8,6)(9,11)(10,13)(12,15)(14,16)(17,18,21,20)(19,22,24,23);(1,4,5,8)(2,6,3,7)(9,10)(11,13)(12,14)(15,16)(17,20,21,18)(19,23,24,22)Polycyclic presentation:
2^9|a^2=gi,b^2=g,c^2=h,e^2=g,f^2=g,b^a=bg,c^a=ch,e^a=ei,c^b=ci,d^b=di,e^c=ei,f^c=fh,e^d=ehi,f^d=fi95 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7988273);