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 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4486 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,6)(3,8)(9,10)(11,15)(12,19)(13,14)(16,20)(17,21)(18,24)(22,23)(25,26,30,33)(27,34,31,28)(29,32,38,40)(35,39,37,36);(2,6)(4,7)(11,17)(15,21)(18,23)(22,24)(25,27)(26,28)(29,37)(30,31)(32,39)(33,34)(35,38)(36,40);(1,2)(3,4)(5,6)(7,8)(10,14)(15,21)(16,19)(22,24)(25,28,30,34)(26,31,33,27)(29,36,38,39)(32,37,40,35);(1,3)(2,7)(4,6)(5,8)(9,10)(11,15)(12,16)(13,14)(17,21)(18,22)(19,20)(23,24)(25,28)(26,27)(29,36)(30,34)(31,33)(32,35)(37,40)(38,39);(1,4,5,7)(2,3,6,8)(9,11,13,17)(10,15,14,21)(12,18,20,23)(16,22,19,24)(25,29)(26,32)(27,35)(28,36)(30,38)(31,37)(33,40)(34,39);(1,3)(2,4)(5,8)(6,7)(9,12,13,20)(10,16,14,19)(11,18,17,23)(15,22,21,24)(25,27)(26,28)(29,37)(30,31)(32,39)(33,34)(35,38)(36,40)Polycyclic presentation:
2^9|a^2=g,c^2=g,e^2=gh,f^2=ghi,b^a=bg,c^a=ch,d^a=di,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, 9621967);