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 | #36318 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4412 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,6,4)(3,7,5,8)(9,10)(11,17)(12,15)(13,16)(14,18)(19,23)(20,22)(21,24)(25,26)(27,36)(28,29)(30,38)(31,32)(33,37)(34,40)(35,39);(1,3,6,5)(2,7,4,8)(9,10,13,16)(11,14,18,17)(12,15,20,22)(19,21,24,23)(26,32)(27,37)(29,34)(35,38);(1,3,6,5)(2,8,4,7)(11,18)(12,20)(14,17)(15,22)(25,27)(26,33)(28,38)(29,39)(30,34)(31,37)(32,36)(35,40);(9,11)(10,14)(12,19)(13,18)(15,21)(16,17)(20,24)(22,23)(25,28)(26,34)(27,38)(29,32)(30,33)(31,40)(35,37)(36,39);(1,4,6,2)(3,7,5,8)(9,12)(10,15)(11,19)(13,20)(14,21)(16,22)(17,23)(18,24)(25,29,31,34)(26,28,32,40)(27,39,37,30)(33,38,36,35);(1,5,6,3)(2,7,4,8)(9,10)(11,14)(12,15)(13,16)(17,18)(19,21)(20,22)(23,24)(25,30,31,39)(26,35,32,38)(27,34,37,29)(28,33,40,36)Polycyclic presentation:
2^9|a^2=hi,b^2=gh,c^2=hi,e^2=h,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, 8679735);