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 | |
4536 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,6,8)(3,7,5,4)(9,10)(11,14)(12,19)(13,15)(16,20)(17,23)(18,21)(22,24)(25,26)(27,34)(28,31)(29,37)(30,33)(32,39)(35,36)(38,40);(1,3,6,5)(2,7,8,4)(9,11,13,18)(10,14,15,21)(12,17,20,24)(16,22,19,23)(25,27)(26,28)(29,35)(30,31)(32,36)(33,34)(37,38)(39,40);(1,2)(3,4)(5,7)(6,8)(10,15)(14,21)(16,19)(22,23)(25,26,30,33)(27,28,31,34)(29,32,39,37)(35,36,40,38);(1,4)(2,5)(3,8)(6,7)(9,10)(11,14)(12,16)(13,15)(17,22)(18,21)(19,20)(23,24)(25,26)(27,34)(28,31)(29,32)(30,33)(35,38)(36,40)(37,39);(1,4)(2,3)(5,8)(6,7)(9,11)(10,14)(12,17)(13,18)(15,21)(16,22)(19,23)(20,24)(25,28,30,34)(26,31,33,27)(29,38,39,36)(32,35,37,40);(1,5,6,3)(2,7,8,4)(9,12,13,20)(10,16,15,19)(11,17,18,24)(14,22,21,23)(25,29,30,39)(26,32,33,37)(27,35,31,40)(28,36,34,38)Polycyclic presentation:
2^9|a^2=i,b^2=gh,c^2=g,e^2=g,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, 9844929);