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 | #52165 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4646 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 32 points:
(9,10,13,11)(12,15,16,14)(17,18)(19,27)(20,30)(21,24)(22,28)(23,25)(26,31)(29,32);(1,2,5,4)(3,6,7,8)(10,11)(14,15)(17,19,23,21)(18,24,25,27)(20,28,31,29)(22,30,32,26);(2,4)(6,8)(9,11,13,10)(12,15,16,14)(18,25)(22,32)(24,27)(26,30);(3,7)(6,8)(9,12)(10,14)(11,15)(13,16)(17,20,23,31)(18,26,25,30)(19,29,21,28)(22,24,32,27);(1,3,5,7)(2,6,4,8)(17,21,23,19)(18,27,25,24)(20,29,31,28)(22,26,32,30);(1,4,5,2)(3,8,7,6)(17,22,23,32)(18,28,25,29)(19,30,21,26)(20,24,31,27)Polycyclic presentation:
2^9|a^2=gh,b^2=gi,c^2=gh,d^2=h,e^2=gi,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^c=fgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8249014);