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 | |
4603 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)(11,18)(12,16)(13,14)(15,17)(19,24)(20,21)(22,23)(25,26,31,34)(27,30,33,40)(28,29,32,39)(35,36,37,38);(1,2,5,4)(3,7,8,6)(9,11,15,12)(10,16,17,18)(13,22,24,20)(14,21,19,23)(25,27)(26,30)(28,37)(29,38)(31,33)(32,35)(34,40)(36,39);(1,3,5,8)(2,6,4,7)(10,17)(14,19)(16,18)(21,23)(27,33)(30,40)(35,37)(36,38);(1,4,5,2)(3,7,8,6)(9,12,15,11)(10,18,17,16)(13,20,24,22)(14,23,19,21)(25,28)(26,29)(27,35)(30,36)(31,32)(33,37)(34,39)(38,40);(9,13)(10,19)(11,20)(12,22)(14,17)(15,24)(16,21)(18,23)(25,29)(26,32)(27,36)(28,34)(30,37)(31,39)(33,38)(35,40);(9,14,15,19)(10,13,17,24)(11,21,12,23)(16,22,18,20)(25,30,31,40)(26,33,34,27)(28,36,32,38)(29,37,39,35)Polycyclic presentation:
2^9|a^2=ghi,b^2=g,c^2=gh,d^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,e^b=egi,f^c=fgi,e^d=egi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8306709);