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 | |
4290 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10,15,14)(11,17,12,16)(13,18,21,22)(19,24,20,23)(25,26)(27,32)(28,29)(30,37)(31,33)(34,36)(35,39)(38,40);(1,2)(3,6)(4,7)(5,8)(9,11,15,12)(10,16,14,17)(13,19,21,20)(18,23,22,24)(25,27)(26,32)(28,36)(29,34)(30,38)(31,35)(33,39)(37,40);(1,3)(2,5)(4,8)(6,7)(26,33)(27,35)(28,38)(34,37);(1,2)(3,5)(4,7)(6,8)(9,12,15,11)(10,17,14,16)(13,19,21,20)(18,23,22,24)(25,28,31,38)(26,29,33,40)(27,36,35,30)(32,34,39,37);(1,3)(2,6)(4,8)(5,7)(9,13,15,21)(10,18,14,22)(11,19,12,20)(16,23,17,24)(25,29)(26,28)(27,37)(30,32)(31,40)(33,38)(34,35)(36,39);(1,2)(3,5)(4,7)(6,8)(9,14,15,10)(11,17,12,16)(13,18,21,22)(19,23,20,24)(25,30,31,36)(26,34,33,37)(27,38,35,28)(29,39,40,32)Polycyclic presentation:
2^9|a^2=g,b^2=g,d^2=gh,e^2=g,f^2=gh,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, 9996178);