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 | #40725 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4510 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)(3,4,8,6)(9,10,14,15)(11,13,17,23)(12,19,21,16)(18,20,24,22)(25,26,30,34)(27,29,33,31)(28,32,36,39)(35,38,40,37);(1,3,5,8)(2,6,7,4)(10,15)(11,17)(16,19)(18,24)(25,27)(26,31)(28,35)(29,34)(30,33)(32,37)(36,40)(38,39);(1,3)(2,4)(5,8)(6,7)(10,15)(11,17)(16,19)(18,24)(28,36)(32,39)(35,40)(37,38);(1,4)(2,3)(5,6)(7,8)(9,11)(10,13)(12,18)(14,17)(15,23)(16,22)(19,20)(21,24)(25,28)(26,32)(27,35)(29,38)(30,36)(31,37)(33,40)(34,39);(9,12,14,21)(10,16,15,19)(11,18,17,24)(13,22,23,20)(25,27)(26,29)(28,35)(30,33)(31,34)(32,38)(36,40)(37,39);(9,13)(10,17)(11,15)(12,20)(14,23)(16,18)(19,24)(21,22)(25,29)(26,33)(27,34)(28,37)(30,31)(32,35)(36,38)(39,40)Polycyclic presentation:
2^9|a^2=g,b^2=i,e^2=h,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8641821);