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 | #38381 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4313 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10,14,15)(11,19,20,16)(12,17,22,21)(13,23,24,18)(25,26)(27,31)(28,33)(29,36)(30,32)(34,40)(35,38)(37,39);(1,2)(3,8)(4,6)(5,7)(10,15)(12,22)(16,19)(18,23)(25,27,30,35)(26,31,32,38)(28,34,39,36)(29,33,40,37);(1,3)(2,4)(5,6)(7,8)(10,15)(11,20)(17,21)(18,23)(26,32)(27,35)(33,37)(34,36);(3,6)(4,8)(9,11,14,20)(10,16,15,19)(12,23,22,18)(13,21,24,17)(25,26)(27,31)(28,37)(29,34)(30,32)(33,39)(35,38)(36,40);(1,4,5,8)(2,6,7,3)(9,12,14,22)(10,17,15,21)(11,18,20,23)(13,16,24,19)(25,28)(26,33)(27,36)(29,31)(30,39)(32,37)(34,35)(38,40);(1,2)(3,4)(5,7)(6,8)(9,13,14,24)(10,18,15,23)(11,21,20,17)(12,19,22,16)(25,29)(26,34)(27,33)(28,38)(30,40)(31,39)(32,36)(35,37)Polycyclic presentation:
2^9|a^2=g,b^2=gh,d^2=g,e^2=hi,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^d=egi,f^e=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9068361);