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 | |
4395 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)(3,5)(4,7)(6,8)(9,10,15,16)(11,13,12,17)(14,18,22,23)(19,21,20,24)(25,26,31,35)(27,32,37,40)(28,29,33,38)(30,39,36,34);(1,3)(2,4)(5,8)(6,7)(10,16)(11,12)(18,23)(19,20)(25,27,31,37)(26,32,35,40)(28,36,33,30)(29,34,38,39);(1,4,6,5)(2,7,8,3)(9,11,15,12)(10,17,16,13)(14,19,22,20)(18,24,23,21)(25,28,31,33)(26,29,35,38)(27,30,37,36)(32,39,40,34);(2,8)(4,5)(9,12,15,11)(10,13,16,17)(14,20,22,19)(18,21,23,24)(25,28)(26,29)(27,36)(30,37)(31,33)(32,34)(35,38)(39,40);(1,5,6,4)(2,3,8,7)(9,13)(10,11)(12,16)(14,21)(15,17)(18,19)(20,23)(22,24)(25,29)(26,33)(27,34)(28,35)(30,32)(31,38)(36,40)(37,39);(1,3)(2,4)(5,8)(6,7)(9,14,15,22)(10,18,16,23)(11,19,12,20)(13,21,17,24)(25,30,31,36)(26,34,35,39)(27,28,37,33)(29,32,38,40)Polycyclic presentation:
2^9|a^2=i,b^2=hi,c^2=ghi,d^2=h,e^2=gh,f^2=i,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8081600);