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 | |
4500 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(3,8)(4,6)(9,10,14,18)(11,15,19,23)(12,21,22,16)(13,20,24,17)(25,26,31,34)(27,35,39,30)(28,32,33,29)(36,40,37,38);(1,2)(3,4)(5,7)(6,8)(9,11)(10,15)(12,17)(13,21)(14,19)(16,24)(18,23)(20,22)(25,27)(26,30)(28,36)(29,40)(31,39)(32,38)(33,37)(34,35);(1,3)(2,6)(4,7)(5,8)(11,19)(12,22)(15,23)(16,21)(25,28,31,33)(26,32,34,29)(27,36,39,37)(30,38,35,40);(1,3,5,8)(2,6,7,4)(9,12)(10,16)(11,20)(13,15)(14,22)(17,19)(18,21)(23,24)(25,28)(26,32)(27,37)(29,34)(30,40)(31,33)(35,38)(36,39);(1,3)(2,4)(5,8)(6,7)(9,13)(10,17)(11,16)(12,23)(14,24)(15,22)(18,20)(19,21)(25,29,31,32)(26,33,34,28)(27,38,39,40)(30,36,35,37);(1,4)(2,3)(5,6)(7,8)(9,11)(10,15)(12,20)(13,16)(14,19)(17,22)(18,23)(21,24)(25,30)(26,27)(28,40)(29,36)(31,35)(32,37)(33,38)(34,39)Polycyclic presentation:
2^9|a^2=ghi,c^2=g,d^2=h,e^2=g,b^a=bg,c^a=ch,d^a=di,e^a=egi,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,d^c=dgi,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, 8841050);