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 | |
4614 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10)(11,17)(12,14)(13,16)(15,19)(18,24)(20,22)(21,23)(25,26,31,35)(27,32,38,37)(28,39,29,33)(30,34,40,36);(1,2)(3,5)(4,6)(7,8)(9,11)(10,15)(12,17)(13,21)(14,19)(16,24)(18,22)(20,23)(25,27)(26,32)(28,34)(29,36)(30,33)(31,38)(35,37)(39,40);(2,3)(6,7)(13,22)(16,20)(18,21)(23,24)(25,28,31,29)(26,33,35,39)(27,36,38,34)(30,32,40,37);(4,8)(6,7)(9,12,14,10)(11,17,19,15)(13,20,22,16)(18,24,21,23)(25,29,31,28)(26,33,35,39)(27,34,38,36)(30,32,40,37);(1,3,5,2)(4,7,8,6)(9,13)(10,16)(11,18)(12,20)(14,22)(15,23)(17,24)(19,21)(25,26)(27,37)(28,39)(29,33)(30,34)(31,35)(32,38)(36,40);(1,4)(2,6)(3,7)(5,8)(9,11)(10,17)(12,15)(13,18)(14,19)(16,24)(20,23)(21,22)(25,30)(26,34)(27,33)(28,32)(29,37)(31,40)(35,36)(38,39)Polycyclic presentation:
2^9|a^2=h,c^2=h,d^2=gh,e^2=hi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,f^d=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8520465);