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 | |
4586 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,5)(6,8)(10,14)(11,17)(16,19)(18,20)(25,26,31,30)(27,34,38,32)(28,33,37,39)(29,35,36,40);(3,7)(6,8)(10,14)(15,22)(16,19)(23,24)(25,27)(26,32)(28,36)(29,37)(30,34)(31,38)(33,35)(39,40);(1,2)(3,6)(4,5)(7,8)(9,10)(11,15)(12,16)(13,14)(17,22)(18,23)(19,21)(20,24)(25,28,31,37)(26,33,30,39)(27,36,38,29)(32,35,34,40);(1,3)(2,6)(4,7)(5,8)(9,10)(11,15)(12,19)(13,14)(16,21)(17,22)(18,24)(20,23)(25,27)(26,34)(28,36)(29,37)(30,32)(31,38)(33,40)(35,39);(1,2)(3,6)(4,5)(7,8)(9,11)(10,15)(12,20)(13,17)(14,22)(16,24)(18,21)(19,23)(25,29,31,36)(26,35,30,40)(27,37,38,28)(32,33,34,39);(9,12,13,21)(10,16,14,19)(11,18,17,20)(15,23,22,24)(25,30,31,26)(27,34,38,32)(28,39,37,33)(29,35,36,40)Polycyclic presentation:
2^9|a^2=g,c^2=g,e^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,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, 8594581);