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 | |
4499 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,4)(6,7)(9,10)(11,16)(12,20)(13,17)(14,19)(15,21)(18,22)(23,24)(25,26)(27,35)(28,37)(29,32)(30,36)(31,33)(34,39)(38,40);(1,2,5,4)(3,7,8,6)(9,11)(10,14)(12,18)(13,19)(15,22)(16,17)(20,23)(21,24)(25,27,31,29)(26,32,33,35)(28,38,39,36)(30,34,40,37);(1,3,5,8)(2,6,4,7)(12,21)(15,20)(18,24)(22,23)(26,33)(28,39)(32,35)(36,38);(1,4,5,2)(3,7,8,6)(9,12)(10,15)(11,18)(13,21)(14,22)(16,23)(17,20)(19,24)(25,28)(26,34)(27,36)(29,38)(30,32)(31,39)(33,37)(35,40);(9,11)(10,16)(12,18)(13,19)(14,17)(15,23)(20,22)(21,24)(25,29,31,27)(26,35,33,32)(28,38,39,36)(30,34,40,37);(9,10)(11,14)(12,20)(13,17)(15,21)(16,19)(18,23)(22,24)(25,30,31,40)(26,36,33,38)(27,34,29,37)(28,35,39,32)Polycyclic presentation:
2^9|b^2=hi,c^2=i,d^2=i,e^2=h,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8627647);