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 | |
4515 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,4,6)(3,5,7,8)(9,10)(11,14)(12,16)(13,17)(15,18)(19,22)(20,21)(23,24)(25,26)(27,37)(28,34)(29,30)(31,36)(32,39)(33,40)(35,38);(3,7)(5,8)(9,11)(10,14)(12,19)(13,18)(15,17)(16,22)(20,23)(21,24)(25,27)(26,32)(28,29)(30,33)(31,39)(34,35)(36,37)(38,40);(1,2)(3,5)(4,6)(7,8)(11,18)(14,15)(19,24)(22,23)(25,28,31,40)(26,33,36,34)(27,29,39,38)(30,37,35,32);(1,3)(2,5)(4,7)(6,8)(9,11)(10,15)(12,19)(13,18)(14,17)(16,23)(20,22)(21,24)(25,28)(26,34)(27,38)(29,39)(30,32)(31,40)(33,36)(35,37);(1,2)(3,5)(4,6)(7,8)(9,12)(10,16)(11,19)(13,21)(14,22)(15,23)(17,20)(18,24)(25,29,31,38)(26,35,36,30)(27,28,39,40)(32,34,37,33);(9,10)(11,14)(12,20)(13,17)(15,18)(16,21)(19,23)(22,24)(25,30,31,35)(26,29,36,38)(27,33,39,34)(28,37,40,32)Polycyclic presentation:
2^9|a^2=gh,c^2=g,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8634015);