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 |
| prime radical | #40436 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4360 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,7)(3,4,6,8)(9,10)(11,13)(12,16)(14,15)(17,21)(18,20)(19,22)(23,24)(25,26,31,33)(27,34,30,32)(28,37,38,29)(35,40,39,36);(1,3,5,6)(2,4,7,8)(12,19)(16,22)(18,23)(20,24)(25,27,31,30)(26,32,33,34)(28,35,38,39)(29,40,37,36);(1,2)(3,8)(4,6)(5,7)(10,15)(11,17)(16,22)(18,23)(26,33)(28,38)(32,34)(35,39);(1,3,5,6)(2,4,7,8)(9,11,14,17)(10,13,15,21)(12,18,19,23)(16,20,22,24)(25,28,31,38)(26,29,33,37)(27,35,30,39)(32,40,34,36);(1,4,5,8)(2,6,7,3)(9,12)(10,16)(11,18)(13,20)(14,19)(15,22)(17,23)(21,24)(25,29)(26,28)(27,36)(30,40)(31,37)(32,39)(33,38)(34,35);(1,2)(3,4)(5,7)(6,8)(9,13)(10,17)(11,15)(12,20)(14,21)(16,23)(18,22)(19,24)(25,30,31,27)(26,34,33,32)(28,39,38,35)(29,36,37,40)Polycyclic presentation:
2^9|a^2=gi,b^2=gi,d^2=h,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, 10060300);