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 | |
5010 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 32 points:
(10,12)(14,16)(17,18)(19,20)(21,30)(22,29)(23,28)(24,25)(26,32)(27,31);(1,2)(3,6)(4,7)(5,8)(11,15)(14,16)(17,19)(18,24)(20,28)(21,27)(22,30)(23,25)(26,31)(29,32);(1,3)(2,5)(4,8)(6,7)(9,10,13,12)(11,14,15,16)(17,20)(18,25)(19,28)(21,31)(22,32)(23,24)(26,27)(29,30);(1,3)(2,6)(4,8)(5,7)(9,11)(10,14)(12,16)(13,15)(17,21,23,32)(18,26,28,30)(19,29,25,27)(20,31,24,22);(9,12,13,10)(11,16,15,14)(17,22,23,31)(18,27,28,29)(19,30,25,26)(20,32,24,21);(1,2)(3,5)(4,7)(6,8)(17,18)(19,24)(20,25)(21,30)(22,27)(23,28)(26,32)(29,31)Polycyclic presentation:
2^9|c^2=gh,d^2=g,e^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^d=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8329360);