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 | |
4488 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,6,8)(3,4,5,7)(9,10)(11,17)(12,15)(13,18)(14,20)(16,21)(19,23)(22,24)(25,26)(27,29)(28,37)(30,32)(31,34)(33,36)(35,39)(38,40);(1,3,6,5)(2,7,8,4)(9,11)(10,14)(12,19)(13,20)(15,22)(16,23)(17,18)(21,24)(25,27)(26,29)(28,38)(30,35)(31,36)(32,39)(33,34)(37,40);(1,3,6,5)(2,7,8,4)(9,12,13,21)(10,15,18,16)(11,19,20,24)(14,22,17,23)(25,28,31,30)(26,32,34,37)(27,35,36,38)(29,40,33,39);(2,8)(3,5)(9,12)(10,16)(11,19)(13,21)(14,23)(15,18)(17,22)(20,24)(25,29,31,33)(26,27,34,36)(28,39,30,40)(32,38,37,35);(1,4)(2,3)(5,8)(6,7)(9,11)(10,17)(12,19)(13,20)(14,18)(15,23)(16,22)(21,24)(25,27)(26,33)(28,35)(29,34)(30,38)(31,36)(32,39)(37,40);(1,5,6,3)(2,7,8,4)(9,10)(11,14)(12,15)(13,18)(16,21)(17,20)(19,22)(23,24)(25,30,31,28)(26,32,34,37)(27,35,36,38)(29,39,33,40)Polycyclic presentation:
2^9|a^2=hi,b^2=hi,c^2=gi,d^2=i,f^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, 8627891);