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 | |
4631 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)(3,4)(10,14)(11,17)(16,21)(18,23)(25,26)(27,34)(28,32)(29,38)(30,33)(31,37)(35,40)(36,39);(1,2)(3,7)(4,8)(5,6)(9,10,13,14)(11,15,17,20)(12,16,19,21)(18,22,23,24)(25,27)(26,31)(28,35)(29,36)(30,37)(32,39)(33,34)(38,40);(1,3,5,4)(2,7,6,8)(9,10)(11,15)(12,16)(13,14)(17,20)(18,22)(19,21)(23,24)(25,28,30,29)(26,32,33,38)(27,35,37,36)(31,39,34,40);(1,2,5,6)(3,7,4,8)(9,10)(11,15)(12,16)(13,14)(17,20)(18,22)(19,21)(23,24)(25,29,30,28)(26,32,33,38)(27,36,37,35)(31,39,34,40);(1,2)(3,7)(4,8)(5,6)(9,11,13,17)(10,15,14,20)(12,18,19,23)(16,22,21,24)(25,26)(27,31)(28,32)(29,38)(30,33)(34,37)(35,39)(36,40);(1,4,5,3)(2,8,6,7)(9,12,13,19)(10,16,14,21)(11,18,17,23)(15,22,20,24)(25,27)(26,31)(28,35)(29,36)(30,37)(32,39)(33,34)(38,40)Polycyclic presentation:
2^9|b^2=i,c^2=gi,d^2=gi,e^2=i,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8055053);