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 | |
4534 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)(11,16)(12,18)(13,19)(14,22)(15,17)(20,24)(21,23)(25,26,31,35)(27,36,39,32)(28,30,38,33)(29,34,37,40);(1,2,5,6)(3,7,8,4)(9,11)(10,16)(12,15)(13,20)(14,21)(17,18)(19,24)(22,23)(25,27)(26,32)(28,37)(29,38)(30,34)(31,39)(33,40)(35,36);(1,2)(3,7)(4,8)(5,6)(10,17)(11,12)(13,23)(14,24)(25,28,31,38)(26,33,35,30)(27,37,39,29)(32,40,36,34);(2,6)(3,8)(9,12,15,11)(10,18,17,16)(13,20,23,22)(14,19,24,21)(25,28)(26,33)(27,29)(30,35)(31,38)(32,34)(36,40)(37,39);(1,3)(2,4)(5,8)(6,7)(9,13,15,23)(10,19,17,21)(11,20,12,22)(14,16,24,18)(25,29)(26,34)(27,38)(28,39)(30,32)(31,37)(33,36)(35,40);(1,4)(2,3)(5,7)(6,8)(9,14)(10,20)(11,21)(12,19)(13,16)(15,24)(17,22)(18,23)(25,30,31,33)(26,28,35,38)(27,34,39,40)(29,32,37,36)Polycyclic presentation:
2^9|a^2=g,b^2=ghi,c^2=g,d^2=gh,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8968629);