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 | |
4816 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 32 points:
(1,2,6,4)(3,7,5,8)(9,10)(11,15)(12,14)(13,16)(17,18,22,23)(19,27,28,24)(20,21,26,25)(29,31,32,30);(1,3,6,5)(2,7,4,8)(9,10,12,14)(11,13,16,15)(18,23)(19,28)(20,26)(30,31);(1,3,6,5)(2,8,4,7)(9,10)(11,13)(12,14)(15,16)(18,23)(21,25)(24,27)(30,31);(1,4,6,2)(3,7,5,8)(9,11)(10,13)(12,16)(14,15)(17,19)(18,24)(20,29)(21,30)(22,28)(23,27)(25,31)(26,32);(1,5,6,3)(2,7,4,8)(17,20,22,26)(18,25,23,21)(19,29,28,32)(24,31,27,30);(17,21,22,25)(18,26,23,20)(19,30,28,31)(24,32,27,29)Polycyclic presentation:
2^9|a^2=h,b^2=gh,c^2=ghi,d^2=ghi,e^2=h,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,f^c=fgi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8154783);