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 | |
4517 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10,14,11)(12,15,19,17)(13,18,21,16)(20,23,24,22)(25,26,31,29)(27,35,36,32)(28,37,38,33)(30,34,40,39);(1,2)(3,6)(4,7)(5,8)(12,19)(15,17)(20,24)(22,23)(25,27)(26,32)(28,30)(29,35)(31,36)(33,34)(37,39)(38,40);(1,3)(2,5)(4,8)(6,7)(10,11)(15,17)(16,18)(22,23)(25,26,31,29)(27,32,36,35)(28,33,38,37)(30,34,40,39);(1,2)(3,5)(4,7)(6,8)(9,11,14,10)(12,17,19,15)(13,18,21,16)(20,23,24,22)(25,28)(26,33)(27,30)(29,37)(31,38)(32,34)(35,39)(36,40);(1,3)(2,6)(4,8)(5,7)(9,12)(10,15)(11,17)(13,20)(14,19)(16,22)(18,23)(21,24)(25,29,31,26)(27,35,36,32)(28,33,38,37)(30,34,40,39);(1,2)(3,5)(4,7)(6,8)(9,13,14,21)(10,16,11,18)(12,20,19,24)(15,22,17,23)(25,30)(26,34)(27,28)(29,39)(31,40)(32,33)(35,37)(36,38)Polycyclic presentation:
2^9|a^2=gh,c^2=g,d^2=h,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^d=egi,f^e=fgi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9996203);