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 | |
4682 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)(11,12)(13,17)(14,15)(16,19)(18,20)(21,22)(23,24)(25,26,30,32)(27,35,37,31)(28,38,39,33)(29,34,40,36);(1,2)(3,4)(5,6)(7,8)(10,15)(12,16)(17,22)(20,23)(25,27,30,37)(26,31,32,35)(28,34,39,36)(29,33,40,38);(2,6)(4,7)(13,21)(17,22)(18,24)(20,23)(26,32)(29,40)(31,35)(33,38);(9,11)(10,16)(12,15)(13,18)(14,19)(17,23)(20,22)(21,24)(25,27)(26,31)(28,36)(29,33)(30,37)(32,35)(34,39)(38,40);(1,3)(2,7)(4,6)(5,8)(9,12,14,16)(10,11,15,19)(13,20,21,23)(17,18,22,24)(25,28,30,39)(26,33,32,38)(27,36,37,34)(29,35,40,31);(1,4,5,7)(2,3,6,8)(9,13,14,21)(10,17,15,22)(11,18,19,24)(12,20,16,23)(25,29)(26,34)(27,33)(28,35)(30,40)(31,39)(32,36)(37,38)Polycyclic presentation:
2^9|a^2=h,b^2=h,e^2=g,f^2=ghi,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, 8139681);