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 | |
4300 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,3)(4,7,8,6)(9,10)(11,16)(12,17)(13,22)(14,15)(18,24)(19,23)(20,21)(25,26)(27,36)(28,32)(29,33)(30,34)(31,35)(37,38)(39,40);(2,3)(6,7)(10,15)(12,21)(16,23)(18,22)(25,27,31,34)(26,30,35,36)(28,37,33,40)(29,38,32,39);(1,3,5,2)(4,7,8,6)(10,15)(11,19)(17,20)(18,22)(25,28,31,33)(26,32,35,29)(27,37,34,40)(30,39,36,38);(1,4)(2,6)(3,7)(5,8)(9,11,14,19)(10,16,15,23)(12,22,21,18)(13,20,24,17)(25,28)(26,32)(27,37)(29,35)(30,39)(31,33)(34,40)(36,38);(9,12,14,21)(10,17,15,20)(11,18,19,22)(13,16,24,23)(25,29,31,32)(26,33,35,28)(27,38,34,39)(30,40,36,37);(9,13,14,24)(10,18,15,22)(11,20,19,17)(12,23,21,16)(25,30)(26,34)(27,35)(28,39)(29,37)(31,36)(32,40)(33,38)Polycyclic presentation:
2^9|a^2=i,b^2=ghi,c^2=gh,d^2=h,e^2=gi,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^d=egi,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9113669);