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 | |
4425 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)(3,4)(5,7)(6,8)(9,10,13,16)(11,17,19,14)(12,15,21,24)(18,23,20,22)(25,26)(27,38)(28,31)(29,39)(30,35)(32,37)(33,36)(34,40);(1,3,6,5)(2,7,8,4)(9,11)(10,14)(12,20)(13,19)(15,23)(16,17)(18,21)(22,24)(25,27)(26,30)(28,39)(29,33)(31,40)(32,35)(34,36)(37,38);(9,11,13,19)(10,14,16,17)(12,18,21,20)(15,22,24,23)(25,28,32,36)(26,33,37,31)(27,39,35,34)(29,38,40,30);(1,4,6,7)(2,5,8,3)(9,11)(10,14)(12,18)(13,19)(15,22)(16,17)(20,21)(23,24)(25,29,32,40)(26,34,37,39)(27,33,35,31)(28,30,36,38);(1,5,6,3)(2,7,8,4)(9,12)(10,15)(11,18)(13,21)(14,22)(16,24)(17,23)(19,20)(25,30)(26,35)(27,37)(28,29)(31,39)(32,38)(33,34)(36,40);(1,2)(3,7)(4,5)(6,8)(9,10)(11,14)(12,15)(13,16)(17,19)(18,22)(20,23)(21,24)(25,31,32,33)(26,36,37,28)(27,40,35,29)(30,34,38,39)Polycyclic presentation:
2^9|a^2=i,b^2=gh,c^2=h,d^2=gi,e^2=gh,f^2=hi,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, 10109107);