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 | |
4445 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)(9,10,14,17)(11,18,12,15)(13,16,21,24)(19,23,20,22)(25,26)(27,32)(28,29)(30,37)(31,33)(34,36)(35,39)(38,40);(2,6)(3,7)(9,11,14,12)(10,15,17,18)(13,19,21,20)(16,22,24,23)(25,27)(26,32)(28,36)(29,34)(30,38)(31,35)(33,39)(37,40);(1,3,5,7)(2,4,6,8)(26,33)(27,35)(28,38)(34,37);(1,4)(2,7)(3,6)(5,8)(9,10)(11,15)(12,18)(13,16)(14,17)(19,22)(20,23)(21,24)(25,28,31,38)(26,29,33,40)(27,36,35,30)(32,34,39,37);(9,12,14,11)(10,15,17,18)(13,19,21,20)(16,23,24,22)(25,29)(26,28)(27,37)(30,32)(31,40)(33,38)(34,35)(36,39);(9,13)(10,16)(11,19)(12,20)(14,21)(15,22)(17,24)(18,23)(25,30,31,36)(26,34,33,37)(27,38,35,28)(29,39,40,32)Polycyclic presentation:
2^9|a^2=ghi,b^2=ghi,c^2=hi,d^2=h,e^2=ghi,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^d=egi,f^e=fgi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10073505);