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 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4304 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10,13,15)(11,16,19,14)(12,20,22,17)(18,23,21,24)(25,26,30,28)(27,31,35,32)(29,36,37,33)(34,39,40,38);(1,2)(3,4)(5,7)(6,8)(9,11,13,19)(10,14,15,16)(12,18,22,21)(17,23,20,24)(25,27)(26,31)(28,32)(29,34)(30,35)(33,38)(36,39)(37,40);(1,3)(2,6)(4,7)(5,8)(10,15)(14,16)(17,20)(23,24)(25,27,30,35)(26,32,28,31)(29,34,37,40)(33,39,36,38);(1,3)(2,4)(5,8)(6,7)(9,10)(11,14)(12,17)(13,15)(16,19)(18,23)(20,22)(21,24)(25,27)(26,31)(28,32)(29,34)(30,35)(33,38)(36,39)(37,40);(1,4,5,6)(2,3,7,8)(9,11)(10,16)(12,21)(13,19)(14,15)(17,23)(18,22)(20,24)(25,28,30,26)(27,31,35,32)(29,33,37,36)(34,39,40,38);(1,3)(2,4)(5,8)(6,7)(9,12)(10,17)(11,18)(13,22)(14,23)(15,20)(16,24)(19,21)(25,29,30,37)(26,33,28,36)(27,34,35,40)(31,38,32,39)Polycyclic presentation:
2^9|a^2=h,b^2=g,c^2=gh,e^2=i,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^d=egi,f^e=fgi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10052769);