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 | #40588 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4976 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 32 points:
(9,10,13,14)(11,12,15,16)(17,18,22,26)(19,27,28,23)(20,24,29,32)(21,25,30,31);(1,2,5,8)(3,4,7,6)(10,14)(11,15)(19,28)(20,29)(23,27)(24,32);(1,3)(2,6)(4,8)(5,7)(9,11)(10,12)(13,15)(14,16)(17,19,22,28)(18,23,26,27)(20,30,29,21)(24,25,32,31);(1,4,5,6)(2,7,8,3)(9,12)(10,11)(13,16)(14,15)(17,20,22,29)(18,24,26,32)(19,21,28,30)(23,31,27,25);(1,3)(2,4)(5,7)(6,8)(17,18)(19,23)(20,24)(21,31)(22,26)(25,30)(27,28)(29,32);(1,4)(2,7)(3,8)(5,6)(17,21)(18,25)(19,20)(22,30)(23,32)(24,27)(26,31)(28,29)Polycyclic presentation:
2^9|a^2=gh,b^2=ghi,c^2=h,d^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8623947);