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 | #37716 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4595 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10)(11,13)(12,20)(14,17)(15,19)(16,21)(18,24)(22,23)(25,26,31,34)(27,35,37,30)(28,38,40,32)(29,39,36,33);(1,2,5,8)(3,4,7,6)(9,11)(10,15)(12,18)(13,17)(14,19)(16,24)(20,22)(21,23)(25,27)(26,30)(28,29)(31,37)(32,39)(33,38)(34,35)(36,40);(1,3)(2,6)(4,8)(5,7)(12,21)(16,20)(18,23)(22,24)(25,28,31,40)(26,32,34,38)(27,29,37,36)(30,39,35,33);(1,4,5,6)(2,7,8,3)(9,12,14,21)(10,16,17,20)(11,18,19,23)(13,22,15,24)(25,28)(26,32)(27,36)(29,37)(30,33)(31,40)(34,38)(35,39);(1,3)(2,4)(5,7)(6,8)(9,13,14,15)(10,11,17,19)(12,22,21,24)(16,18,20,23)(25,29,31,36)(26,33,34,39)(27,28,37,40)(30,38,35,32);(1,4)(2,7)(3,8)(5,6)(9,11)(10,13)(12,23)(14,19)(15,17)(16,24)(18,21)(20,22)(25,30)(26,27)(28,39)(29,32)(31,35)(33,40)(34,37)(36,38)Polycyclic presentation:
2^9|a^2=h,b^2=i,c^2=h,d^2=ghi,e^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8623989);