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 | |
4702 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,4,6)(3,5,7,8)(9,10,14,15)(11,19,20,16)(12,22,23,17)(13,18,24,21)(25,26)(27,32)(28,37)(29,33)(30,36)(31,35)(34,40)(38,39);(3,7)(5,8)(10,15)(11,20)(17,22)(18,21)(25,27,31,38)(26,32,35,39)(28,34,29,36)(30,37,40,33);(1,2)(3,5)(4,6)(7,8)(10,15)(12,23)(16,19)(18,21)(25,28,31,29)(26,33,35,37)(27,36,38,34)(30,32,40,39);(1,3,4,7)(2,5,6,8)(9,11)(10,16)(12,21)(13,17)(14,20)(15,19)(18,23)(22,24)(25,29,31,28)(26,33,35,37)(27,34,38,36)(30,32,40,39);(1,2)(3,5)(4,6)(7,8)(9,12)(10,17)(11,21)(13,16)(14,23)(15,22)(18,20)(19,24)(25,30)(26,34)(27,37)(28,39)(29,32)(31,40)(33,38)(35,36);(9,13)(10,18)(11,22)(12,16)(14,24)(15,21)(17,20)(19,23)(25,26)(27,32)(28,33)(29,37)(30,34)(31,35)(36,40)(38,39)Polycyclic presentation:
2^9|a^2=hi,b^2=i,c^2=i,d^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^c=egi,f^d=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8439317);