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 | #47457 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4300 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,6,5)(3,8,4,7)(9,10)(11,14)(12,17)(13,18)(15,16)(19,23)(20,22)(21,24)(25,26)(27,33)(28,35)(29,32)(30,31)(34,40)(36,37)(38,39);(1,3,6,4)(2,7,5,8)(9,11)(10,14)(12,13)(15,19)(16,23)(17,18)(20,21)(22,24)(25,27)(26,28)(29,37)(30,35)(31,33)(32,40)(34,38)(36,39);(10,16)(11,19)(12,20)(18,24)(26,31)(28,33)(32,39)(36,40);(1,3,6,4)(2,7,5,8)(9,12,15,20)(10,17,16,22)(11,13,19,21)(14,18,23,24)(25,28)(26,27)(29,36)(30,33)(31,35)(32,34)(37,39)(38,40);(1,4,6,3)(2,7,5,8)(9,13)(10,18)(11,20)(12,19)(14,22)(15,21)(16,24)(17,23)(25,29)(26,32)(27,34)(28,36)(30,38)(31,39)(33,40)(35,37);(1,5,6,2)(3,8,4,7)(9,14,15,23)(10,19,16,11)(12,18,20,24)(13,17,21,22)(25,27)(26,28)(29,37)(30,35)(31,33)(32,40)(34,38)(36,39)Polycyclic presentation:
2^9|a^2=ghi,b^2=ghi,d^2=gh,e^2=ghi,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, 9613488);