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 | |
5085 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 32 points:
(10,13)(14,16)(17,18)(19,26)(20,30)(21,25)(22,23)(24,29)(27,32)(28,31);(1,2,5,6)(3,7,8,4)(11,15)(14,16)(18,23)(20,25)(24,26)(27,31);(2,6)(4,7)(9,10)(11,14)(12,13)(15,16)(18,23)(21,30)(24,26)(28,32);(2,6)(3,8)(9,11,12,15)(10,14,13,16)(17,19,22,29)(18,24,23,26)(20,31,25,27)(21,28,30,32);(1,3)(2,4)(5,8)(6,7)(9,10)(11,14)(12,13)(15,16)(17,20,22,25)(18,21,23,30)(19,27,29,31)(24,32,26,28);(1,4)(2,3)(5,7)(6,8)(17,21)(18,25)(19,28)(20,23)(22,30)(24,27)(26,31)(29,32)Polycyclic presentation:
2^9|b^2=i,d^2=h,e^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,e^b=egi,f^c=fgi,e^d=egi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8294955);