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 | #47830 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4628 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)(11,14)(12,16)(13,17)(15,18)(19,22)(20,21)(23,24)(25,26,30,32)(27,28,34,36)(29,31,40,39)(33,35,37,38);(1,3,5,8)(2,6,7,4)(9,11)(10,14)(12,19)(13,18)(15,17)(16,22)(20,23)(21,24)(25,27)(26,28)(29,37)(30,34)(31,38)(32,36)(33,40)(35,39);(1,3)(2,4)(5,8)(6,7)(11,18)(14,15)(19,24)(22,23)(25,28,30,36)(26,27,32,34)(29,35,40,38)(31,33,39,37);(1,4)(2,3)(5,6)(7,8)(9,11)(10,15)(12,19)(13,18)(14,17)(16,23)(20,22)(21,24)(25,28)(26,27)(29,38)(30,36)(31,37)(32,34)(33,39)(35,40);(9,10)(11,14)(12,20)(13,17)(15,18)(16,21)(19,23)(22,24)(25,29)(26,31)(27,33)(28,35)(30,40)(32,39)(34,37)(36,38);(9,12)(10,16)(11,19)(13,21)(14,22)(15,23)(17,20)(18,24)(25,26)(27,28)(29,39)(30,32)(31,40)(33,38)(34,36)(35,37)Polycyclic presentation:
2^9|a^2=gh,b^2=g,c^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^d=egi,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9073952);