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 | |
4616 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,4)(6,7)(9,10)(11,13)(12,16)(14,15)(17,21)(18,20)(19,22)(23,24)(25,26)(27,30)(28,33)(29,34)(31,32)(35,40)(36,38)(37,39);(1,2,5,4)(3,7,8,6)(12,19)(16,22)(18,23)(20,24)(25,27,31,35)(26,30,32,40)(28,36,37,29)(33,38,39,34);(1,3,5,8)(2,6,4,7)(10,15)(11,17)(16,22)(18,23)(26,32)(27,35)(28,37)(34,38);(1,4,5,2)(3,7,8,6)(9,11,14,17)(10,13,15,21)(12,18,19,23)(16,20,22,24)(25,28)(26,33)(27,36)(29,35)(30,38)(31,37)(32,39)(34,40);(9,12)(10,16)(11,18)(13,20)(14,19)(15,22)(17,23)(21,24)(25,29,31,36)(26,34,32,38)(27,37,35,28)(30,39,40,33);(9,13)(10,17)(11,15)(12,20)(14,21)(16,23)(18,22)(19,24)(25,30)(26,35)(27,32)(28,38)(29,33)(31,40)(34,37)(36,39)Polycyclic presentation:
2^9|b^2=i,c^2=g,d^2=hi,e^2=gi,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, 10078849);