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 | |
4845 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 32 points:
(1,2)(3,4)(5,6)(7,8)(10,12)(14,15)(17,18,22,25)(19,20,27,23)(21,24,30,29)(26,28,32,31);(1,3,7,6)(2,4,8,5)(9,10,13,12)(11,15,16,14)(17,19,22,27)(18,23,25,20)(21,26,30,32)(24,31,29,28);(1,4,7,5)(2,6,8,3)(9,11,13,16)(10,14,12,15);(1,4,7,5)(2,3,8,6)(9,12,13,10)(11,15,16,14)(17,18)(19,23)(20,27)(21,29)(22,25)(24,30)(26,28)(31,32);(1,5,7,4)(2,3,8,6)(17,20,22,23)(18,19,25,27)(21,28,30,31)(24,26,29,32);(1,6,7,3)(2,4,8,5)(17,21)(18,24)(19,26)(20,28)(22,30)(23,31)(25,29)(27,32)Polycyclic presentation:
2^9|a^2=ghi,b^2=gh,c^2=i,d^2=i,e^2=gi,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=egi,e^d=eghi,f^d=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10352138);