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 | |
4658 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)(11,15)(12,20)(13,14)(16,21)(17,19)(18,23)(22,24)(25,26)(27,34)(28,33)(29,30)(31,35)(32,38)(36,37)(39,40);(1,2)(3,4)(5,6)(7,8)(9,10)(11,17)(12,20)(13,14)(15,19)(16,21)(18,22)(23,24)(25,27)(26,32)(28,36)(29,37)(30,39)(31,38)(33,40)(34,35);(2,6)(4,7)(10,14)(15,17)(16,20)(22,23)(25,28,31,30)(26,33,35,29)(27,36,38,39)(32,40,34,37);(9,11)(10,15)(12,18)(13,19)(14,17)(16,22)(20,23)(21,24)(25,29)(26,28)(27,37)(30,35)(31,33)(32,36)(34,39)(38,40);(1,3)(2,7)(4,6)(5,8)(9,12)(10,16)(11,18)(13,21)(14,20)(15,22)(17,23)(19,24)(25,30,31,28)(26,29,35,33)(27,36,38,39)(32,40,34,37);(1,4,5,7)(2,3,6,8)(9,10)(11,15)(12,16)(13,14)(17,19)(18,22)(20,21)(23,24)(25,27)(26,34)(28,39)(29,37)(30,36)(31,38)(32,35)(33,40)Polycyclic presentation:
2^9|c^2=g,e^2=g,f^2=hi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,f^c=fgi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8139461);