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 | |
4547 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)(3,4)(5,6)(7,8)(9,10,14,15)(11,19,22,16)(12,17,18,13)(20,24,23,21);(2,6)(3,7)(10,15)(12,18)(16,19)(20,23)(25,26,30,31)(27,32,37,40)(28,33,38,36)(29,34,39,35);(1,3,5,7)(2,4,6,8)(10,15)(13,17)(16,19)(21,24)(26,31)(29,39)(32,40)(33,36);(1,4)(2,7)(3,6)(5,8)(9,11,14,22)(10,16,15,19)(12,23,18,20)(13,21,17,24)(25,27)(26,32)(28,34)(29,36)(30,37)(31,40)(33,39)(35,38);(9,12,14,18)(10,17,15,13)(11,20,22,23)(16,21,19,24)(25,28)(26,33)(27,35)(29,32)(30,38)(31,36)(34,37)(39,40);(9,13,14,17)(10,18,15,12)(11,21,22,24)(16,20,19,23)(25,29)(26,34)(27,36)(28,32)(30,39)(31,35)(33,37)(38,40)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,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, 8805225);