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 |
| prime radical | #47874 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4570 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)(3,5)(4,6)(7,8)(9,10)(11,15)(12,13)(14,16)(18,19)(23,25)(24,26)(31,32);(1,3,7,6)(2,4,8,5)(10,13)(14,15)(17,18,22,19)(20,23,28,25)(21,24,30,26)(27,31,29,32);(1,4,7,5)(2,3,8,6)(9,10)(11,14)(12,13)(15,16);(2,8)(4,5)(9,11)(10,14)(12,16)(13,15)(17,19,22,18)(20,23,28,25)(21,26,30,24)(27,31,29,32);(1,5,7,4)(2,3,8,6)(17,20,22,28)(18,23,19,25)(21,29,30,27)(24,32,26,31);(1,6,7,3)(2,5,8,4)(17,21,22,30)(18,24,19,26)(20,27,28,29)(23,31,25,32)Polycyclic presentation:
2^9|b^2=gi,c^2=h,d^2=ghi,e^2=gi,f^2=gi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^d=egi,f^e=fgi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10123798);