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 | |
5289 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 6 generators on 32 points:
(11,15)(13,16)(17,18,22,23)(19,25,26,24)(20,29,30,21)(27,32,31,28);(1,2)(3,7)(4,5)(6,8)(9,10,12,14)(11,13,15,16)(18,23)(19,26)(20,30)(28,32);(2,5)(6,7)(9,11)(10,13)(12,15)(14,16)(18,23)(21,29)(24,25)(28,32);(3,8)(6,7)(9,10)(11,13)(12,14)(15,16)(17,19,22,26)(18,24,23,25)(20,31,30,27)(21,28,29,32);(1,3)(2,6)(4,8)(5,7)(9,11)(10,13)(12,15)(14,16)(17,20)(18,21)(19,27)(22,30)(23,29)(24,32)(25,28)(26,31);(1,2)(3,6)(4,5)(7,8)(17,21)(18,20)(19,28)(22,29)(23,30)(24,31)(25,27)(26,32)Polycyclic presentation:
2^9|a^2=g,b^2=gh,d^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,f^c=fgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8253752);