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 | |
4464 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 32 points:
(1,2,5,4)(3,7,8,6)(10,13)(11,12)(17,18,21,24)(19,22,25,29)(20,27,28,23)(26,32,31,30);(1,3)(2,6)(4,7)(5,8)(9,10,14,13)(11,16,12,15)(19,25)(22,29)(26,31)(30,32);(9,11,14,12)(10,15,13,16)(17,18)(19,22)(20,23)(21,24)(25,29)(26,30)(27,28)(31,32);(1,3)(2,7)(4,6)(5,8)(9,12,14,11)(10,16,13,15)(17,19,21,25)(18,22,24,29)(20,26,28,31)(23,30,27,32);(1,3)(2,7)(4,6)(5,8)(9,13,14,10)(11,16,12,15)(17,20,21,28)(18,23,24,27)(19,26,25,31)(22,30,29,32);(1,4,5,2)(3,7,8,6)Polycyclic presentation:
2^9|a^2=gh,b^2=i,c^2=i,d^2=h,e^2=h,f^2=gi,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,f^d=fgi,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8595410);