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 | |
4387 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10,13,14)(11,15,19,17)(12,16,21,22)(18,23,20,24)(25,26,30,31)(27,29,33,37)(28,32,35,38)(34,36,39,40);(1,2)(3,7)(4,6)(5,8)(10,14)(15,17)(16,22)(23,24)(28,35)(32,38)(34,39)(36,40);(1,3)(2,5)(4,8)(6,7)(26,31)(27,33)(32,38)(34,39);(2,6)(5,7)(9,10)(11,17)(12,16)(13,14)(15,19)(18,24)(20,23)(21,22)(25,27)(26,29)(28,34)(30,33)(31,37)(32,36)(35,39)(38,40);(1,2)(3,5)(4,6)(7,8)(9,11)(10,15)(12,20)(13,19)(14,17)(16,24)(18,21)(22,23)(25,28)(26,32)(27,34)(29,36)(30,35)(31,38)(33,39)(37,40);(1,3)(2,7)(4,8)(5,6)(9,12,13,21)(10,16,14,22)(11,18,19,20)(15,23,17,24)(25,29)(26,27)(28,36)(30,37)(31,33)(32,34)(35,40)(38,39)Polycyclic presentation:
2^9|a^2=gh,f^2=g,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, 10124999);