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 | |
4476 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10,14,15)(11,19,20,16)(12,17,22,21)(13,23,24,18)(25,26)(27,31)(28,32)(29,30)(33,35)(34,39)(36,38)(37,40);(1,2,5,6)(3,7,8,4)(10,15)(11,20)(17,21)(18,23)(25,27,30,33)(26,31,29,35)(28,34,36,37)(32,39,38,40);(1,2)(3,7)(4,8)(5,6)(10,15)(11,20)(12,22)(13,24)(26,29)(27,33)(28,36)(39,40);(2,6)(3,8)(9,11,14,20)(10,16,15,19)(12,23,22,18)(13,21,24,17)(25,27)(26,31)(28,37)(29,35)(30,33)(32,40)(34,36)(38,39);(1,3)(2,4)(5,8)(6,7)(9,12)(10,17)(11,18)(13,19)(14,22)(15,21)(16,24)(20,23)(25,28)(26,32)(27,34)(29,38)(30,36)(31,39)(33,37)(35,40);(1,4)(2,3)(5,7)(6,8)(9,13,14,24)(10,18,15,23)(11,21,20,17)(12,19,22,16)(25,29,30,26)(27,35,33,31)(28,38,36,32)(34,40,37,39)Polycyclic presentation:
2^9|a^2=g,b^2=i,d^2=g,f^2=h,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, 8968678);