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 | |
4652 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,5)(3,4)(9,10,14,12)(11,17,19,15)(13,16,21,20)(18,23,24,22)(25,26,30,34)(27,31,35,40)(28,32,38,37)(29,33,39,36);(1,2,6,5)(3,8,4,7)(10,12)(15,17)(16,20)(22,23)(27,35)(28,38)(31,40)(32,37);(1,3,6,4)(2,7,5,8)(13,21)(16,20)(18,24)(22,23)(25,27)(26,31)(28,36)(29,32)(30,35)(33,38)(34,40)(37,39);(1,4,6,3)(2,8,5,7)(9,11)(10,15)(12,17)(13,18)(14,19)(16,22)(20,23)(21,24);(1,5,6,2)(3,8,4,7)(9,12,14,10)(11,17,19,15)(13,20,21,16)(18,23,24,22)(25,28)(26,32)(27,36)(29,31)(30,38)(33,35)(34,37)(39,40);(9,13)(10,16)(11,18)(12,20)(14,21)(15,22)(17,23)(19,24)(25,29)(26,33)(27,37)(28,31)(30,39)(32,35)(34,36)(38,40)Polycyclic presentation:
2^9|a^2=gi,b^2=h,c^2=h,d^2=h,e^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,f^c=fgi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8109634);