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 | |
4547 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)(11,14)(12,20)(13,16)(15,22)(17,19)(18,24)(21,23)(25,26)(27,36)(28,39)(29,30)(31,32)(33,37)(34,38)(35,40);(1,2)(3,7)(4,5)(6,8)(9,11,13,19)(10,14,16,17)(12,21,22,18)(15,24,20,23)(26,32)(27,37)(29,35)(34,39);(2,5)(6,7)(9,10,13,16)(11,17,19,14)(12,15,22,20)(18,24,21,23)(25,27)(26,33)(28,35)(29,38)(30,39)(31,37)(32,36)(34,40);(1,2)(3,6)(4,5)(7,8)(9,10)(11,14)(12,15)(13,16)(17,19)(18,23)(20,22)(21,24)(25,28,31,38)(26,34,32,39)(27,35,37,29)(30,33,40,36);(1,2)(3,6)(4,5)(7,8)(9,11)(10,14)(12,18)(13,19)(15,23)(16,17)(20,24)(21,22)(25,29,31,35)(26,30,32,40)(27,38,37,28)(33,39,36,34);(1,3)(2,6)(4,8)(5,7)(9,12)(10,15)(11,18)(13,22)(14,23)(16,20)(17,24)(19,21)(25,30)(26,35)(27,39)(28,33)(29,32)(31,40)(34,37)(36,38)Polycyclic presentation:
2^9|b^2=gh,c^2=gh,d^2=g,e^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^c=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8665255);