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 | #35569 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4484 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 32 points:
(1,2,6,7)(3,8,4,5)(9,10)(11,12)(13,14)(15,16);(1,3,6,4)(2,5,7,8)(10,14)(11,15)(17,18)(19,24)(20,25)(21,26)(22,23)(27,31)(28,29)(30,32);(2,7)(5,8)(9,11,13,15)(10,12,14,16)(18,23)(21,31)(24,29)(25,32);(2,7)(3,4)(9,12)(10,15)(11,14)(13,16)(17,19)(18,24)(20,26)(21,25)(22,28)(23,29)(27,30)(31,32);(1,4,6,3)(2,8,7,5)(17,20)(18,25)(19,27)(21,29)(22,30)(23,32)(24,31)(26,28);(1,5,6,8)(2,3,7,4)(17,21,22,31)(18,26,23,27)(19,25,28,32)(20,29,30,24)Polycyclic presentation:
2^9|a^2=h,b^2=h,c^2=gh,e^2=h,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, 8229582);