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 | #58312 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4198 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10,15,14)(11,18,16,12)(13,21,22,17)(19,20,23,24)(26,28)(31,33)(32,36)(38,39);(1,2)(3,7)(4,5)(6,8)(9,11,15,16)(10,12,14,18)(13,19,22,23)(17,20,21,24)(25,26,30,28)(27,33,35,31)(29,32,37,36)(34,39,40,38);(2,5)(6,7)(9,12,15,18)(10,16,14,11)(13,20,22,24)(17,23,21,19)(25,27,30,35)(26,31,28,33)(29,34,37,40)(32,38,36,39);(9,11)(10,12)(13,19)(14,18)(15,16)(17,20)(21,24)(22,23)(25,28,30,26)(27,33,35,31)(29,36,37,32)(34,39,40,38);(1,3,4,8)(2,6,5,7)(9,13,15,22)(10,17,14,21)(11,19,16,23)(12,20,18,24);(1,2)(3,6)(4,5)(7,8)(9,14,15,10)(11,18,16,12)(13,21,22,17)(19,24,23,20)(25,29,30,37)(26,32,28,36)(27,34,35,40)(31,38,33,39)Polycyclic presentation:
2^9|a^2=h,b^2=g,c^2=g,d^2=gh,e^2=ghi,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,f^c=fgi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8121967);