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 | #40490 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
5046 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 32 points:
(2,4)(6,8)(9,10)(11,12)(13,14)(15,16)(17,18)(19,22)(20,31)(21,32)(23,25)(24,30)(26,29)(27,28);(3,7)(6,8)(10,14)(11,15)(17,19,23,30)(18,24,25,22)(20,28,29,21)(26,27,31,32);(1,2,5,4)(3,6,7,8)(9,11,13,15)(10,12,14,16)(18,25)(22,24)(26,31)(27,32);(1,3,5,7)(2,6,4,8)(9,12)(10,15)(11,14)(13,16)(17,20)(18,26)(19,28)(21,30)(22,32)(23,29)(24,27)(25,31);(1,4,5,2)(3,8,7,6)(17,21,23,28)(18,27,25,32)(19,29,30,20)(22,31,24,26);(17,22)(18,19)(20,32)(21,31)(23,24)(25,30)(26,28)(27,29)Polycyclic presentation:
2^9|b^2=gi,c^2=gh,d^2=ghi,e^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,e^b=egi,f^c=fgi,e^d=egi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8300976);