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 | |
4847 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 32 points:
(9,10)(11,15)(12,14)(13,16)(17,18)(19,22)(20,23)(21,26)(24,29)(25,27)(28,31)(30,32);(1,2)(3,5)(4,7)(6,8)(9,10,12,14)(11,13,16,15)(18,19)(23,25)(24,26)(30,31);(1,3)(2,7)(4,6)(5,8)(9,10)(11,13)(12,14)(15,16)(18,19)(20,27)(21,29)(30,31);(2,8)(3,4)(9,11,12,16)(10,13,14,15)(17,19,22,18)(20,23,27,25)(21,26,29,24)(28,30,32,31);(1,4,6,3)(2,5,8,7)(17,20)(18,23)(19,25)(21,28)(22,27)(24,30)(26,31)(29,32);(1,5)(2,3)(4,8)(6,7)(17,21,22,29)(18,24,19,26)(20,28,27,32)(23,30,25,31)Polycyclic presentation:
2^9|b^2=gh,d^2=g,e^2=ghi,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,d^c=dgi,e^c=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, 8970433);