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 | |
4770 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 32 points:
(10,13)(14,16)(17,18,22,23)(19,27,28,24)(20,25,30,29)(21,31,32,26);(1,2)(3,8)(4,6)(5,7)(11,15)(14,16)(18,23)(20,30)(24,27)(26,31);(1,3,5,4)(2,6,7,8)(9,10)(11,14)(12,13)(15,16)(18,23)(19,28)(25,29)(26,31);(3,4)(6,8)(17,19,22,28)(18,24,23,27)(20,31,30,26)(21,29,32,25);(1,4,5,3)(2,6,7,8)(9,11)(10,14)(12,15)(13,16)(17,20,22,30)(18,25,23,29)(19,26,28,31)(21,24,32,27);(1,2)(3,6)(4,8)(5,7)(9,10)(11,14)(12,13)(15,16)(17,21,22,32)(18,26,23,31)(19,29,28,25)(20,27,30,24)Polycyclic presentation:
2^9|a^2=g,c^2=i,d^2=g,e^2=gi,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^d=egi,f^e=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9071242);