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 | |
4398 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)(3,4)(5,6)(7,8)(9,10,15,17)(11,16,20,13)(12,23,22,18)(14,24,21,19)(25,26,30,32)(27,28,33,35)(29,31,38,40)(34,36,37,39);(2,6)(3,7)(9,11,15,20)(10,16,17,13)(12,14,22,21)(18,19,23,24)(27,33)(28,35)(34,37)(36,39);(1,3,5,7)(2,4,6,8)(10,17)(11,20)(12,22)(19,24)(25,27)(26,28)(29,34)(30,33)(31,36)(32,35)(37,38)(39,40);(1,4)(2,7)(3,6)(5,8)(9,12,15,22)(10,18,17,23)(11,21,20,14)(13,19,16,24)(25,28)(26,27)(29,36)(30,35)(31,34)(32,33)(37,40)(38,39);(9,13)(10,11)(12,19)(14,23)(15,16)(17,20)(18,21)(22,24)(25,27)(26,28)(29,37)(30,33)(31,39)(32,35)(34,38)(36,40);(9,14,15,21)(10,19,17,24)(11,22,20,12)(13,18,16,23)(25,29)(26,31)(27,34)(28,36)(30,38)(32,40)(33,37)(35,39)Polycyclic presentation:
2^9|a^2=gi,b^2=h,c^2=g,d^2=h,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,d^c=dh,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9845326);