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 | |
4894 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 32 points:
(9,10)(11,13)(12,15)(14,16)(17,18,23,26)(19,24,29,22)(20,21,32,25)(27,31,30,28);(1,2,5,8)(3,4,7,6)(10,11)(14,15)(17,19,23,29)(18,24,26,22)(20,30,32,27)(21,28,25,31);(1,3)(2,6)(4,8)(5,7)(9,10,13,11)(12,14,16,15)(17,20,23,32)(18,25,26,21)(19,27,29,30)(22,31,24,28);(1,4,5,6)(2,7,8,3)(9,11,13,10)(12,15,16,14)(17,20)(18,21)(19,27)(22,28)(23,32)(24,31)(25,26)(29,30);(1,3)(2,4)(5,7)(6,8)(9,12,13,16)(10,14,11,15)(17,21)(18,20)(19,28)(22,27)(23,25)(24,30)(26,32)(29,31);(1,4)(2,7)(3,8)(5,6)(17,22)(18,19)(20,31)(21,30)(23,24)(25,27)(26,29)(28,32)Polycyclic presentation:
2^9|a^2=gh,b^2=gi,c^2=h,d^2=ghi,e^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,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, 8623992);