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 |
| prime radical | #59117 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4756 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 32 points:
(11,12)(14,15)(17,18,22,23)(19,27,28,24)(20,25,31,30)(21,26,32,29);(1,2)(3,7)(4,8)(5,6)(9,10,13,16)(11,14,12,15)(18,23)(19,28)(25,30)(26,29);(1,3)(2,4)(5,8)(6,7)(9,11,13,12)(10,14,16,15);(2,6)(4,7)(9,10)(11,14)(12,15)(13,16)(17,19,22,28)(18,24,23,27)(20,29,31,26)(21,25,32,30);(1,3)(2,7)(4,6)(5,8)(9,12,13,11)(10,15,16,14)(17,20)(18,25)(19,29)(21,27)(22,31)(23,30)(24,32)(26,28);(1,4,5,7)(2,8,6,3)(17,21,22,32)(18,26,23,29)(19,30,28,25)(20,24,31,27)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=h,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,f^d=fgi,f^e=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8606468);