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 | |
4372 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)(11,12)(13,16)(14,15)(17,19)(18,20)(21,22)(23,24)(25,26,31,28)(27,30,35,32)(29,33,38,36)(34,37,40,39);(1,2,4,6)(3,5,8,7)(10,15)(12,19)(16,22)(20,24)(25,26,31,28)(27,32,35,30)(29,36,38,33)(34,37,40,39);(1,2)(3,5)(4,6)(7,8)(11,17)(12,19)(13,21)(16,22)(25,27,31,35)(26,32,28,30)(29,34,38,40)(33,39,36,37);(1,3,4,8)(2,5,6,7)(9,11,14,17)(10,12,15,19)(13,18,21,23)(16,20,22,24)(25,28,31,26)(27,30,35,32)(29,36,38,33)(34,37,40,39);(9,12)(10,11)(13,20)(14,19)(15,17)(16,18)(21,24)(22,23)(25,29)(26,33)(27,34)(28,36)(30,37)(31,38)(32,39)(35,40);(9,13,14,21)(10,16,15,22)(11,18,17,23)(12,20,19,24)(25,30)(26,27)(28,35)(29,37)(31,32)(33,34)(36,40)(38,39)Polycyclic presentation:
2^9|a^2=h,b^2=hi,c^2=h,d^2=gh,f^2=gi,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8964568);