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 | |
4520 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)(3,7)(4,6)(5,8)(9,10,14,15)(11,12,17,16)(13,21,22,18)(19,23,24,20)(25,26,31,35)(27,32,36,30)(28,33,37,40)(29,34,38,39);(1,2,4,6)(3,5,8,7)(10,15)(11,17)(18,21)(19,24)(25,27,31,36)(26,32,35,30)(28,38,37,29)(33,39,40,34);(1,2)(3,5)(4,6)(7,8)(10,15)(12,16)(13,22)(19,24)(27,36)(29,38)(30,32)(34,39);(1,3,4,8)(2,5,6,7)(9,11)(10,16)(12,15)(13,19)(14,17)(18,23)(20,21)(22,24)(25,28)(26,33)(27,29)(30,39)(31,37)(32,34)(35,40)(36,38);(9,12,14,16)(10,17,15,11)(13,20,22,23)(18,24,21,19)(25,29)(26,34)(27,37)(28,36)(30,33)(31,38)(32,40)(35,39);(9,13)(10,18)(11,19)(12,20)(14,22)(15,21)(16,23)(17,24)(25,30)(26,27)(28,39)(29,33)(31,32)(34,37)(35,36)(38,40)Polycyclic presentation:
2^9|a^2=gi,b^2=i,d^2=gh,e^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,e^c=egi,f^c=fgi,e^d=egi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8948191);