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 | |
5082 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 32 points:
(2,5)(6,8)(10,12)(14,15)(17,18)(19,24)(20,26)(21,30)(22,29)(23,25)(27,31)(28,32);(3,7)(6,8)(9,10,13,12)(11,15,16,14)(17,19)(18,24)(20,23)(21,29)(22,30)(25,26)(27,32)(28,31);(1,2)(3,6)(4,5)(7,8)(9,11,13,16)(10,14,12,15)(18,25)(19,20)(27,30)(28,29);(1,3)(2,6)(4,7)(5,8)(9,12,13,10)(11,15,16,14)(17,20,23,19)(18,26,25,24)(21,29,31,28)(22,27,32,30);(1,2)(3,6)(4,5)(7,8)(17,21)(18,27)(19,29)(20,28)(22,26)(23,31)(24,32)(25,30);(17,22)(18,28)(19,30)(20,27)(21,26)(23,32)(24,31)(25,29)Polycyclic presentation:
2^9|b^2=g,c^2=g,d^2=i,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8066503);