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 | |
4380 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)(11,12)(13,21)(14,16)(15,17)(18,22)(19,23)(20,24)(25,26,30,35)(27,33,36,31)(28,32,34,29)(37,38,39,40);(1,2,5,4)(3,7,8,6)(9,11,14,17)(10,15,16,12)(13,19,22,24)(18,23,21,20)(27,36)(31,33)(37,39)(38,40);(1,3,5,8)(2,6,4,7)(10,16)(12,15)(13,22)(19,24)(25,27,30,36)(26,31,35,33)(28,39,34,37)(29,40,32,38);(3,8)(6,7)(9,12,14,15)(10,17,16,11)(13,20,22,23)(18,24,21,19)(25,28,30,34)(26,32,35,29)(27,37,36,39)(31,40,33,38);(1,3)(2,7)(4,6)(5,8)(9,10)(11,12)(13,18)(14,16)(15,17)(19,20)(21,22)(23,24)(25,27)(26,33)(28,39)(29,38)(30,36)(31,35)(32,40)(34,37);(1,4,5,2)(3,6,8,7)(9,13,14,22)(10,18,16,21)(11,19,17,24)(12,20,15,23)(25,29,30,32)(26,34,35,28)(27,38,36,40)(31,39,33,37)Polycyclic presentation:
2^9|a^2=gh,b^2=hi,c^2=i,d^2=h,f^2=gi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8935357);