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 | |
4478 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(4,8)(6,7)(9,10)(11,16)(12,18)(13,23)(14,20)(15,17)(19,21)(22,24)(26,29)(27,35)(28,38)(34,40);(1,2,5,3)(4,6,8,7)(9,11,15,20)(10,16,17,14)(12,21,22,13)(18,19,24,23)(25,26,31,29)(27,36,35,32)(28,39,38,33)(30,34,37,40);(2,3)(6,7)(10,17)(11,20)(13,21)(18,24)(25,27)(26,32)(28,37)(29,36)(30,38)(31,35)(33,40)(34,39);(9,12)(10,18)(11,13)(14,19)(15,22)(16,23)(17,24)(20,21)(25,28,31,38)(26,33,29,39)(27,37,35,30)(32,40,36,34);(1,3,5,2)(4,7,8,6)(9,13)(10,19)(11,12)(14,18)(15,21)(16,24)(17,23)(20,22)(25,29,31,26)(27,36,35,32)(28,39,38,33)(30,40,37,34);(1,4)(2,6)(3,7)(5,8)(9,14)(10,11)(12,19)(13,18)(15,16)(17,20)(21,24)(22,23)(25,30,31,37)(26,34,29,40)(27,38,35,28)(32,39,36,33)Polycyclic presentation:
2^9|b^2=i,d^2=g,e^2=ghi,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^c=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8027308);