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 | |
4326 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10,14,15)(11,18,17,13)(12,16,20,24)(19,22,21,23)(25,26)(27,35)(28,32)(29,33)(30,34)(31,37)(36,40)(38,39);(1,2)(3,7)(4,6)(5,8)(10,15)(11,17)(12,20)(22,23)(25,27,30,37)(26,31,34,35)(28,29,36,39)(32,38,40,33);(1,3)(2,5)(4,8)(6,7)(10,15)(11,17)(16,24)(19,21)(25,28,30,36)(26,32,34,40)(27,29,37,39)(31,38,35,33);(2,6)(5,7)(9,11,14,17)(10,13,15,18)(12,21,20,19)(16,22,24,23)(25,27)(26,31)(28,29)(30,37)(32,38)(33,40)(34,35)(36,39);(1,2)(3,5)(4,6)(7,8)(9,12)(10,16)(11,19)(13,23)(14,20)(15,24)(17,21)(18,22)(25,29)(26,33)(27,36)(28,37)(30,39)(31,32)(34,38)(35,40);(1,3)(2,7)(4,8)(5,6)(9,13)(10,17)(11,15)(12,22)(14,18)(16,19)(20,23)(21,24)(25,26)(27,31)(28,32)(29,38)(30,34)(33,39)(35,37)(36,40)Polycyclic presentation:
2^9|a^2=h,b^2=gh,c^2=gh,d^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,d^c=dh,e^d=egi,f^e=fgi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10125858);