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 | |
4533 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,6)(4,8)(9,10)(11,15)(12,13)(14,16)(17,21)(18,19)(20,22)(23,24)(25,26)(27,36)(28,33)(29,40)(30,34)(31,35)(32,39)(37,38);(2,6)(3,7)(9,11,14,20)(10,15,16,22)(12,18,17,23)(13,19,21,24)(25,27,31,39)(26,32,35,36)(28,37,40,34)(29,38,33,30);(1,2)(3,4)(5,6)(7,8)(10,16)(12,17)(15,22)(18,23)(25,28,31,40)(26,33,35,29)(27,37,39,34)(30,36,38,32);(2,6)(3,7)(9,12,14,17)(10,13,16,21)(11,18,20,23)(15,19,22,24)(25,29)(26,28)(27,38)(30,39)(31,33)(32,37)(34,36)(35,40);(1,3,5,7)(2,4,6,8)(9,11)(10,15)(12,18)(13,19)(14,20)(16,22)(17,23)(21,24)(25,26)(27,32)(28,33)(29,40)(30,37)(31,35)(34,38)(36,39);(1,4)(2,3)(5,8)(6,7)(9,13)(10,17)(11,19)(12,16)(14,21)(15,23)(18,22)(20,24)(25,30,31,38)(26,34,35,37)(27,29,39,33)(28,36,40,32)Polycyclic presentation:
2^9|b^2=ghi,c^2=g,d^2=hi,e^2=i,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^d=egi,f^e=fgi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10078937);