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 | |
4179 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,6)(7,8)(9,10,14,15)(11,16,19,12)(13,21,22,17)(18,20,24,23)(26,31)(28,34)(32,33)(35,40);(1,2,5,6)(3,7,4,8)(10,15)(12,16)(17,21)(20,23)(25,26,30,31)(27,33,37,32)(28,29,34,39)(35,38,40,36);(9,11,14,19)(10,12,15,16)(13,18,22,24)(17,20,21,23)(25,27)(26,32)(28,35)(29,36)(30,37)(31,33)(34,40)(38,39);(1,3,5,4)(2,7,6,8)(9,11)(10,16)(12,15)(13,18)(14,19)(17,23)(20,21)(22,24)(25,28,30,34)(26,29,31,39)(27,35,37,40)(32,36,33,38);(1,3,5,4)(2,7,6,8)(9,12,14,16)(10,11,15,19)(13,20,22,23)(17,18,21,24)(25,27)(26,33)(28,35)(29,38)(30,37)(31,32)(34,40)(36,39);(1,4,5,3)(2,8,6,7)(9,13)(10,17)(11,18)(12,20)(14,22)(15,21)(16,23)(19,24)(25,29)(26,34)(27,36)(28,31)(30,39)(32,40)(33,35)(37,38)Polycyclic presentation:
2^9|a^2=h,b^2=gh,c^2=h,d^2=gh,e^2=gi,f^2=ghi,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dh,f^e=fi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7965134);