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 | |
5769 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 6 generators on 32 points:
(11,16)(13,14)(17,18,23,26)(19,27,29,22)(20,30,32,24)(21,31,28,25);(1,2)(3,6)(4,7)(5,8)(9,10)(11,14)(12,15)(13,16)(17,19)(18,22)(20,21)(23,29)(24,31)(25,30)(26,27)(28,32);(1,3)(2,5)(4,8)(6,7)(9,11)(10,13)(12,16)(14,15)(17,20,23,32)(18,24,26,30)(19,21,29,28)(22,31,27,25);(3,8)(5,6)(9,10,12,15)(11,14,16,13)(17,20)(18,24)(19,28)(21,29)(22,25)(23,32)(26,30)(27,31);(1,2)(3,5)(4,7)(6,8)(9,11)(10,14)(12,16)(13,15)(17,21,23,28)(18,25,26,31)(19,20,29,32)(22,30,27,24);(1,3)(2,6)(4,8)(5,7)(9,10)(11,13)(12,15)(14,16)(17,22)(18,19)(20,31)(21,24)(23,27)(25,32)(26,29)(28,30)Polycyclic presentation:
2^9|a^2=g,c^2=g,d^2=gh,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, 8609453);