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 | |
4858 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 32 points:
(1,2,5,3)(4,6,8,7)(17,18,23,24)(19,25,20,22)(21,29,31,26)(27,30,28,32);(1,2)(3,5)(4,6)(7,8)(9,10)(11,15)(12,13)(14,16)(18,24)(19,20)(26,29)(27,28);(4,8)(6,7)(10,13)(14,15)(17,19,23,20)(18,22,24,25)(21,27,31,28)(26,30,29,32);(2,3)(6,7)(9,11,12,16)(10,14,13,15)(17,20,23,19)(18,25,24,22)(21,28,31,27)(26,32,29,30);(1,3,5,2)(4,7,8,6)(9,11)(10,14)(12,16)(13,15)(17,21,23,31)(18,26,24,29)(19,27,20,28)(22,30,25,32);(1,4,5,8)(2,6,3,7)(9,10)(11,14)(12,13)(15,16)(17,22,23,25)(18,19,24,20)(21,30,31,32)(26,27,29,28)Polycyclic presentation:
2^9|a^2=g,c^2=h,d^2=i,e^2=g,f^2=g,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, 8298112);