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 | |
4918 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 32 points:
(2,5)(6,8)(9,10,13,11)(12,15,16,14)(17,18)(19,24)(20,21)(22,30)(23,25)(26,31)(27,29)(28,32);(3,7)(6,8)(10,11)(14,15)(17,19)(18,24)(20,23)(21,25)(22,27)(26,32)(28,31)(29,30);(1,2)(3,6)(4,5)(7,8)(9,11,13,10)(12,15,16,14)(18,25)(19,20)(26,30)(27,28);(1,3)(2,6)(4,7)(5,8)(9,12)(10,14)(11,15)(13,16)(17,20,23,19)(18,21,25,24)(22,28,31,27)(26,29,30,32);(1,2)(3,6)(4,5)(7,8)(17,21)(18,19)(20,25)(22,29)(23,24)(26,27)(28,30)(31,32);(17,22)(18,26)(19,27)(20,28)(21,29)(23,31)(24,32)(25,30)Polycyclic presentation:
2^9|a^2=g,c^2=g,d^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^b=egi,d^c=dgi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8019666);