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 | |
4405 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2)(3,4)(5,6)(7,8)(9,10,14,12)(11,17,19,15)(13,20,21,16)(18,22,24,23);(2,6)(3,7)(10,12)(15,17)(16,20)(22,23)(25,26,30,31)(27,28,32,35)(29,33,37,38)(34,36,39,40);(1,3,5,7)(2,4,6,8)(10,12)(13,21)(15,17)(18,24)(26,31)(28,35)(29,37)(34,39);(1,4)(2,7)(3,6)(5,8)(9,11,14,19)(10,15,12,17)(13,18,21,24)(16,22,20,23)(25,27)(26,28)(29,34)(30,32)(31,35)(33,36)(37,39)(38,40);(9,12,14,10)(11,15,19,17)(13,20,21,16)(18,22,24,23)(25,28)(26,32)(27,31)(29,36)(30,35)(33,39)(34,38)(37,40);(9,13,14,21)(10,16,12,20)(11,18,19,24)(15,22,17,23)(25,29)(26,33)(27,34)(28,36)(30,37)(31,38)(32,39)(35,40)Polycyclic presentation:
2^9|a^2=h,b^2=ghi,c^2=gh,d^2=h,e^2=h,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,e^c=egi,f^c=fgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8849816);