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 | |
4373 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)(11,17,18,12)(13,21,22,16)(19,20,24,23)(25,26)(27,33)(28,30)(29,39)(31,34)(32,38)(35,40)(36,37);(1,2,5,4)(3,7,8,6)(9,11,14,18)(10,12,15,17)(13,19,22,24)(16,20,21,23)(25,27)(26,32)(28,35)(29,36)(30,37)(31,38)(33,34)(39,40);(1,3,5,8)(2,6,4,7)(10,15)(11,18)(16,21)(19,24)(25,28,31,29)(26,30,34,39)(27,35,38,36)(32,37,33,40);(1,3,5,8)(2,7,4,6)(9,10)(11,12)(13,16)(14,15)(17,18)(19,20)(21,22)(23,24)(25,29,31,28)(26,30,34,39)(27,36,38,35)(32,37,33,40);(1,3)(2,7)(4,6)(5,8)(9,12)(10,11)(13,20)(14,17)(15,18)(16,19)(21,24)(22,23)(25,27)(26,33)(28,36)(29,35)(30,37)(31,38)(32,34)(39,40);(1,4,5,2)(3,6,8,7)(9,13)(10,16)(11,19)(12,20)(14,22)(15,21)(17,23)(18,24)(25,30)(26,28)(27,37)(29,34)(31,39)(32,35)(33,36)(38,40)Polycyclic presentation:
2^9|a^2=h,b^2=i,c^2=gi,d^2=gi,f^2=hi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^c=egi,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, 9514402);