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 | |
4537 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,6)(3,4,7,8)(9,10)(11,19)(12,17)(13,21)(14,15)(16,20)(18,24)(22,23)(25,26)(27,36)(28,33)(29,38)(30,34)(31,35)(32,37)(39,40);(2,6)(3,7)(10,15)(12,23)(16,19)(18,21)(25,27)(26,32)(28,34)(29,30)(31,37)(33,40)(35,36)(38,39);(1,3)(2,4)(5,7)(6,8)(10,15)(11,20)(17,22)(18,21)(25,28,31,38)(26,33,35,29)(27,34,37,39)(30,32,40,36);(1,4)(2,3)(5,8)(6,7)(9,11,14,20)(10,16,15,19)(12,18,23,21)(13,22,24,17)(25,29)(26,28)(27,30)(31,33)(32,34)(35,38)(36,39)(37,40);(9,12)(10,17)(11,21)(13,19)(14,23)(15,22)(16,24)(18,20)(25,26)(27,32)(28,33)(29,38)(30,39)(31,35)(34,40)(36,37);(9,13,14,24)(10,18,15,21)(11,22,20,17)(12,16,23,19)(25,30)(26,34)(27,29)(28,32)(31,40)(33,37)(35,39)(36,38)Polycyclic presentation:
2^9|a^2=i,c^2=ghi,d^2=h,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, 10015852);