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 | |
4611 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,7)(3,4,8,6)(9,10)(11,16)(12,20)(13,17)(14,19)(15,21)(18,22)(23,24)(25,26,31,35)(27,36,32,28)(29,38,40,33)(30,34,39,37);(1,3,5,8)(2,6,7,4)(9,11)(10,14)(12,18)(13,19)(15,22)(16,17)(20,23)(21,24)(25,27)(26,28)(29,34)(30,38)(31,32)(33,39)(35,36)(37,40);(1,3)(2,4)(5,8)(6,7)(12,21)(15,20)(18,24)(22,23)(25,28,31,36)(26,32,35,27)(29,39,40,30)(33,37,38,34);(1,4)(2,3)(5,6)(7,8)(9,12)(10,15)(11,18)(13,21)(14,22)(16,23)(17,20)(19,24)(25,29,31,40)(26,33,35,38)(27,37,32,34)(28,30,36,39);(9,11)(10,16)(12,18)(13,19)(14,17)(15,23)(20,22)(21,24)(25,27)(26,28)(29,37)(30,33)(31,32)(34,40)(35,36)(38,39);(9,10)(11,14)(12,20)(13,17)(15,21)(16,19)(18,23)(22,24)(25,30)(26,34)(27,38)(28,29)(31,39)(32,33)(35,37)(36,40)Polycyclic presentation:
2^9|a^2=hi,b^2=i,c^2=h,d^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8619486);