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 | |
4794 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10,14,15)(11,18,17,13)(12,16,20,24)(19,22,21,23)(25,26)(27,34)(28,36)(29,31)(30,33)(32,38)(35,39)(37,40);(1,2,5,6)(3,7,8,4)(10,15)(11,17)(12,20)(22,23)(25,27,30,29)(26,31,33,34)(28,37,38,35)(32,40,36,39);(1,3)(2,4)(5,8)(6,7)(10,15)(11,17)(16,24)(19,21);(2,6)(4,7)(9,11,14,17)(10,13,15,18)(12,21,20,19)(16,22,24,23)(25,28,30,38)(26,32,33,36)(27,35,29,37)(31,39,34,40);(1,4)(2,3)(5,7)(6,8)(9,12,14,20)(10,16,15,24)(11,19,17,21)(13,23,18,22)(25,29,30,27)(26,31,33,34)(28,35,38,37)(32,40,36,39);(1,3)(2,7)(4,6)(5,8)(9,13)(10,17)(11,15)(12,22)(14,18)(16,19)(20,23)(21,24)(25,26)(27,31)(28,32)(29,34)(30,33)(35,39)(36,38)(37,40)Polycyclic presentation:
2^9|a^2=h,b^2=ghi,d^2=g,e^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,d^c=dh,e^d=egi,f^e=fgi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10132787);