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 | |
4556 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)(11,19)(12,17)(13,18)(14,15)(16,20)(21,22)(23,24)(25,26,31,35)(27,32,37,39)(28,30,38,33)(29,34,36,40);(1,2)(3,7)(4,6)(5,8)(10,15)(11,20)(17,22)(18,24)(25,27)(26,32)(28,29)(30,34)(31,37)(33,40)(35,39)(36,38);(1,3)(2,5)(4,8)(6,7)(25,28)(26,33)(27,36)(29,37)(30,35)(31,38)(32,34)(39,40);(2,6)(5,7)(9,11)(10,16)(12,18)(13,22)(14,20)(15,19)(17,23)(21,24)(25,29,31,36)(26,34,35,40)(27,28,37,38)(30,39,33,32);(1,3)(2,7)(4,8)(5,6)(9,12)(10,17)(11,18)(13,19)(14,21)(15,22)(16,23)(20,24)(25,30,31,33)(26,28,35,38)(27,34,37,40)(29,39,36,32);(1,2)(3,5)(4,6)(7,8)(9,13,14,23)(10,18,15,24)(11,17,20,22)(12,16,21,19)(25,28)(26,30)(27,29)(31,38)(32,34)(33,35)(36,37)(39,40)Polycyclic presentation:
2^9|a^2=h,d^2=h,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,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, 8597011);