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 | |
4828 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 32 points:
(10,12)(14,16)(17,18,23,22)(19,25,20,24)(21,29,30,26)(27,31,28,32);(1,2)(3,4)(5,6)(7,8)(11,15)(14,16)(17,19,23,20)(18,24,22,25)(21,28,30,27)(26,32,29,31);(2,6)(4,7)(9,10,13,12)(11,14,15,16)(18,22)(24,25)(26,29)(31,32);(2,6)(3,8)(9,11)(10,14)(12,16)(13,15)(17,20,23,19)(18,25,22,24)(21,27,30,28)(26,31,29,32);(1,3)(2,7)(4,6)(5,8)(9,12,13,10)(11,16,15,14)(17,21,23,30)(18,26,22,29)(19,27,20,28)(24,31,25,32);(1,4,5,7)(2,3,6,8)(17,22,23,18)(19,25,20,24)(21,29,30,26)(27,32,28,31)Polycyclic presentation:
2^9|a^2=g,b^2=g,c^2=gh,d^2=g,e^2=h,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,e^b=egi,f^c=fgi,e^d=egi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8306275);