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 | |
4882 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 32 points:
(9,10,13,11)(12,15,16,14)(17,18)(19,27)(20,29)(21,30)(22,26)(23,24)(25,28)(31,32);(1,2,5,8)(3,4,7,6)(10,11)(14,15)(17,19,23,29)(18,20,24,27)(21,22,28,31)(25,26,30,32);(1,3)(2,6)(4,8)(5,7)(9,11,13,10)(12,15,16,14)(18,24)(20,27)(25,30)(26,32);(1,4,5,6)(2,7,8,3)(9,12)(10,14)(11,15)(13,16)(17,20)(18,19)(21,26)(22,25)(23,27)(24,29)(28,32)(30,31);(1,3)(2,4)(5,7)(6,8)(17,21)(18,25)(19,22)(20,26)(23,28)(24,30)(27,32)(29,31);(1,4)(2,7)(3,8)(5,6)(17,22)(18,26)(19,28)(20,30)(21,29)(23,31)(24,32)(25,27)Polycyclic presentation:
2^9|a^2=gh,b^2=gi,c^2=gh,d^2=ghi,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, 8623986);