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 | #34981 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
5308 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 6 generators on 32 points:
(2,4)(6,8)(9,10,13,14)(11,12,15,16)(17,18)(19,24)(20,22)(21,30)(23,25)(26,32)(27,31)(28,29);(3,7)(6,8)(10,14)(11,15)(17,19,23,21)(18,24,25,30)(20,29,26,27)(22,28,32,31);(1,2,5,4)(3,6,7,8)(9,11)(10,12)(13,15)(14,16)(18,25)(19,21)(20,26)(28,31);(1,3,5,7)(2,6,4,8)(9,12)(10,11)(13,16)(14,15)(17,20,23,26)(18,22,25,32)(19,27,21,29)(24,31,30,28);(1,4,5,2)(3,8,7,6)(17,21,23,19)(18,24,25,30)(20,29,26,27)(22,31,32,28);(17,22)(18,26)(19,28)(20,25)(21,31)(23,32)(24,27)(29,30)Polycyclic presentation:
2^9|a^2=g,b^2=gi,c^2=ghi,d^2=h,e^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi,e^c=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8034701);