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 | #50641 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4240 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10,15,17)(11,16,14,24)(12,18,21,13)(19,20,22,23)(25,26)(27,34)(28,37)(29,31)(30,33)(32,36)(35,39)(38,40);(1,2)(3,8)(4,6)(5,7)(9,11,15,14)(10,16,17,24)(12,19,21,22)(13,23,18,20)(25,27)(26,32)(28,35)(29,34)(30,38)(31,36)(33,39)(37,40);(1,3,5,6)(2,4,7,8)(10,17)(12,21)(16,24)(19,22)(25,28,31,30)(26,33,29,37)(27,35,36,38)(32,39,34,40);(9,12)(10,18)(11,19)(13,17)(14,22)(15,21)(16,20)(23,24)(25,29,31,26)(27,34,36,32)(28,33,30,37)(35,39,38,40);(1,2)(3,4)(5,7)(6,8)(9,13,15,18)(10,12,17,21)(11,20,14,23)(16,22,24,19)(25,30,31,28)(26,33,29,37)(27,35,36,38)(32,40,34,39);(1,4)(2,6)(3,7)(5,8)(9,14,15,11)(10,16,17,24)(12,22,21,19)(13,23,18,20)(25,27)(26,32)(28,35)(29,34)(30,38)(31,36)(33,39)(37,40)Polycyclic presentation:
2^9|a^2=gh,b^2=gh,c^2=gi,d^2=g,e^2=h,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9591211);