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 | #50857 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4228 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10,13,16)(11,14,18,22)(12,19,20,15)(17,23,24,21)(26,31)(27,29)(33,36)(35,37);(1,2)(3,7)(4,5)(6,8)(9,11,13,18)(10,14,16,22)(12,17,20,24)(15,21,19,23)(25,26,30,31)(27,32,29,34)(28,36,38,33)(35,40,37,39);(2,5)(6,7)(9,10)(11,14)(12,15)(13,16)(17,21)(18,22)(19,20)(23,24)(25,26)(27,32)(28,33)(29,34)(30,31)(35,39)(36,38)(37,40);(1,2)(3,6)(4,5)(7,8)(25,27,30,29)(26,32,31,34)(28,35,38,37)(33,39,36,40);(1,2)(3,6)(4,5)(7,8)(9,11)(10,14)(12,17)(13,18)(15,21)(16,22)(19,23)(20,24)(25,28)(26,33)(27,35)(29,37)(30,38)(31,36)(32,39)(34,40);(1,3)(2,6)(4,8)(5,7)(9,12,13,20)(10,15,16,19)(11,17,18,24)(14,21,22,23)(25,29,30,27)(26,34,31,32)(28,37,38,35)(33,40,36,39)Polycyclic presentation:
2^9|a^2=gh,b^2=h,d^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^c=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8667070);