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 | |
4554 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,6)(3,4,7,8)(9,10,14,11)(12,15,19,17)(13,16,21,18)(20,22,24,23)(25,26,30,31)(27,32,37,39)(28,29,34,33)(35,40,38,36);(2,6)(3,7)(10,11)(12,19)(16,18)(20,24)(26,31)(28,34)(32,39)(35,38);(1,3)(2,4)(5,7)(6,8)(13,21)(16,18)(20,24)(22,23)(26,31)(29,33)(32,39)(36,40);(1,4)(2,3)(5,8)(6,7)(9,11,14,10)(12,15,19,17)(13,18,21,16)(20,22,24,23)(25,27)(26,32)(28,38)(29,36)(30,37)(31,39)(33,40)(34,35);(9,12,14,19)(10,15,11,17)(13,20,21,24)(16,22,18,23)(25,28,30,34)(26,33,31,29)(27,35,37,38)(32,36,39,40);(9,13)(10,16)(11,18)(12,20)(14,21)(15,22)(17,23)(19,24)(25,29,30,33)(26,34,31,28)(27,36,37,40)(32,35,39,38)Polycyclic presentation:
2^9|a^2=g,d^2=ghi,e^2=gi,f^2=h,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dghi,e^b=egi,f^c=fgi,e^d=egi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8324027);