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 | |
4528 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,4)(3,7,8,6)(10,14)(11,17)(16,21)(18,23)(25,26)(27,28)(29,37)(30,32)(31,39)(33,35)(34,40)(36,38);(1,3,5,8)(2,6,4,7)(9,10,13,14)(11,15,17,20)(12,16,19,21)(18,22,23,24)(25,27,30,35)(26,28,32,33)(29,38,39,34)(31,40,37,36);(2,4)(6,7)(9,10)(11,15)(12,16)(13,14)(17,20)(18,22)(19,21)(23,24)(25,26,30,32)(27,33,35,28)(29,31,39,37)(34,40,38,36);(1,3)(2,6)(4,7)(5,8)(9,10)(11,15)(12,16)(13,14)(17,20)(18,22)(19,21)(23,24)(25,28)(26,27)(29,36)(30,33)(31,34)(32,35)(37,38)(39,40);(1,3)(2,6)(4,7)(5,8)(9,11)(10,15)(12,18)(13,17)(14,20)(16,22)(19,23)(21,24)(25,29)(26,31)(27,34)(28,36)(30,39)(32,37)(33,40)(35,38);(1,4,5,2)(3,7,8,6)(9,12,13,19)(10,16,14,21)(11,18,17,23)(15,22,20,24)(25,26)(27,28)(29,31)(30,32)(33,35)(34,36)(37,39)(38,40)Polycyclic presentation:
2^9|a^2=hi,b^2=h,c^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,f^c=fgi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8109205);