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 | #2703 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4083 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,4)(6,7)(9,10,15,17)(11,20,14,16)(12,18,23,22)(13,19,24,21)(25,26)(27,33)(28,34)(29,37)(30,31)(32,38)(35,36)(39,40);(1,2,5,4)(3,7,8,6)(9,11,15,14)(10,16,17,20)(12,21,23,19)(13,18,24,22)(26,31)(27,34)(32,37)(35,40);(1,3,5,8)(2,6,4,7)(10,17)(12,23)(16,20)(19,21)(25,27)(26,28)(29,35)(30,34)(31,33)(32,36)(37,39)(38,40);(1,4,5,2)(3,7,8,6)(9,12)(10,18)(11,21)(13,20)(14,19)(15,23)(16,24)(17,22)(25,28)(26,27)(29,36)(30,33)(31,34)(32,35)(37,40)(38,39);(9,13,15,24)(10,19,17,21)(11,22,14,18)(12,16,23,20)(25,26)(27,28)(29,32)(30,31)(33,34)(35,36)(37,38)(39,40);(9,14,15,11)(10,20,17,16)(12,19,23,21)(13,22,24,18)(25,29,30,38)(26,32,31,37)(27,35,34,40)(28,36,33,39)Polycyclic presentation:
2^9|a^2=gi,b^2=hi,c^2=gh,d^2=gh,e^2=gi,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^d=egi,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9113672);