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 | |
4896 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 32 points:
(3,6)(4,8)(9,10)(11,15)(12,14)(13,16)(17,18,23,27)(19,28,31,24)(20,29,30,25)(21,22,26,32);(1,2)(3,8)(4,6)(5,7)(9,10,12,14)(11,13,16,15)(17,19,23,31)(18,24,27,28)(20,22,30,32)(21,25,26,29);(1,3,5,6)(2,4,7,8)(9,10)(11,13)(12,14)(15,16);(1,2)(3,8)(4,6)(5,7)(9,11,12,16)(10,13,14,15)(17,20,23,30)(18,25,27,29)(19,22,31,32)(21,24,26,28);(1,2)(3,4)(5,7)(6,8)(17,21)(18,22)(19,29)(20,24)(23,26)(25,31)(27,32)(28,30);(1,4)(2,6)(3,7)(5,8)(17,22)(18,26)(19,30)(20,31)(21,27)(23,32)(24,29)(25,28)Polycyclic presentation:
2^9|a^2=ghi,b^2=g,c^2=h,d^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eghi,d^c=dh,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9690071);