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 | |
4383 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,4)(6,8)(9,10,14,17)(11,15,19,13)(12,16,21,24)(18,20,23,22)(25,26,30,32)(27,31,36,40)(28,29,37,39)(33,34,35,38);(9,11,14,19)(10,13,17,15)(12,18,21,23)(16,22,24,20)(25,27)(26,31)(28,35)(29,38)(30,36)(32,40)(33,37)(34,39);(1,2,5,4)(3,6,7,8)(25,28)(26,29)(27,33)(30,37)(31,34)(32,39)(35,36)(38,40);(3,7)(6,8)(9,11)(10,15)(12,18)(13,17)(14,19)(16,20)(21,23)(22,24)(25,29)(26,28)(27,34)(30,39)(31,33)(32,37)(35,40)(36,38);(1,3)(2,6)(4,8)(5,7)(9,12)(10,16)(11,18)(13,22)(14,21)(15,20)(17,24)(19,23)(25,28)(26,29)(27,35)(30,37)(31,38)(32,39)(33,36)(34,40);(1,4,5,2)(3,8,7,6)(9,13,14,15)(10,11,17,19)(12,20,21,22)(16,23,24,18)(25,27)(26,31)(28,33)(29,34)(30,36)(32,40)(35,37)(38,39)Polycyclic presentation:
2^9|a^2=gi,b^2=g,c^2=h,f^2=gh,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^d=eh,f^e=fgi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10365164);