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 | |
4748 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 32 points:
(1,2,5,8)(3,4,7,6)(9,10,13,14)(11,12,15,16)(17,18,22,23)(19,24,27,30)(20,25,26,21)(28,31,32,29);(1,3)(2,6)(4,8)(5,7)(10,14)(11,15)(18,23)(19,27)(20,26)(29,31);(1,4)(2,7)(3,8)(5,6)(9,11)(10,12)(13,15)(14,16)(18,23)(19,27)(20,26)(29,31);(1,2,5,8)(3,4,7,6)(9,12)(10,11)(13,16)(14,15)(17,19)(18,24)(20,28)(21,29)(22,27)(23,30)(25,31)(26,32);(1,3)(2,4)(5,7)(6,8)(17,20)(18,25)(19,28)(21,23)(22,26)(24,31)(27,32)(29,30);(1,4)(2,7)(3,8)(5,6)(17,21,22,25)(18,26,23,20)(19,29,27,31)(24,32,30,28)Polycyclic presentation:
2^9|a^2=g,d^2=ghi,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,d^c=dh,e^c=egi,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9945691);