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 | #56769 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4213 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,6,5)(3,8,4,7)(9,10,14,11)(12,15,20,17)(13,18,22,16)(19,24,21,23)(26,31)(27,33)(29,36)(34,39);(1,3,6,4)(2,7,5,8)(9,10,14,11)(12,17,20,15)(13,16,22,18)(19,24,21,23)(25,26,30,31)(27,32,33,38)(28,36,37,29)(34,40,39,35);(10,11)(15,17)(16,18)(23,24)(25,27,30,33)(26,32,31,38)(28,34,37,39)(29,35,36,40);(1,3,6,4)(2,7,5,8)(9,11,14,10)(12,17,20,15)(13,18,22,16)(19,24,21,23)(25,27)(26,32)(28,34)(29,35)(30,33)(31,38)(36,40)(37,39);(1,4,6,3)(2,7,5,8)(9,12)(10,15)(11,17)(13,21)(14,20)(16,24)(18,23)(19,22)(25,28,30,37)(26,29,31,36)(27,34,33,39)(32,35,38,40);(1,5,6,2)(3,8,4,7)(9,13)(10,16)(11,18)(12,19)(14,22)(15,23)(17,24)(20,21)(25,29)(26,28)(27,35)(30,36)(31,37)(32,34)(33,40)(38,39)Polycyclic presentation:
2^9|a^2=gh,b^2=gi,c^2=hi,d^2=gh,e^2=g,f^2=ghi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9619361);