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 | |
4349 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)(9,10,14,15)(11,16,19,23)(12,17,21,20)(13,18,24,22)(25,26,31,34)(27,32,35,40)(28,38,36,30)(29,33,39,37);(1,2,6,8)(3,5,7,4)(10,15)(12,21)(16,23)(18,22)(25,27)(26,32)(28,29)(30,37)(31,35)(33,38)(34,40)(36,39);(1,3,6,7)(2,4,8,5)(11,19)(13,24)(16,23)(18,22)(27,35)(28,36)(30,38)(32,40);(1,4,6,5)(2,3,8,7)(9,11,14,19)(10,16,15,23)(12,22,21,18)(13,20,24,17)(25,28)(26,30)(27,29)(31,36)(32,37)(33,40)(34,38)(35,39);(1,2)(3,4)(5,7)(6,8)(9,12)(10,17)(11,18)(13,23)(14,21)(15,20)(16,24)(19,22)(25,29)(26,33)(27,36)(28,35)(30,32)(31,39)(34,37)(38,40);(1,5,6,4)(2,7,8,3)(9,13,14,24)(10,18,15,22)(11,20,19,17)(12,23,21,16)(25,30)(26,28)(27,37)(29,32)(31,38)(33,35)(34,36)(39,40)Polycyclic presentation:
2^9|a^2=gi,b^2=h,c^2=h,d^2=g,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dgi,f^c=fgi,e^d=egi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8835928);