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 | |
4270 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2)(3,7)(4,6)(5,8)(9,10,13,16)(11,14,19,17)(12,15,21,24)(18,22,20,23)(25,26)(27,32)(28,30)(29,34)(31,33)(35,40)(36,39)(37,38);(1,2,4,6)(3,5,8,7)(9,10)(11,14)(12,15)(13,16)(17,19)(18,22)(20,23)(21,24)(25,27)(26,32)(28,37)(29,35)(30,38)(31,36)(33,39)(34,40);(1,2)(3,5)(4,6)(7,8)(26,33)(27,36)(28,35)(34,38);(1,3,4,8)(2,5,6,7)(9,10)(11,17)(12,15)(13,16)(14,19)(18,23)(20,22)(21,24)(25,28,31,35)(26,30,33,40)(27,37,36,29)(32,38,39,34);(9,11,13,19)(10,14,16,17)(12,20,21,18)(15,23,24,22)(25,29,31,37)(26,34,33,38)(27,28,36,35)(30,39,40,32);(9,12,13,21)(10,15,16,24)(11,18,19,20)(14,22,17,23)(25,30)(26,35)(27,38)(28,33)(29,32)(31,40)(34,36)(37,39)Polycyclic presentation:
2^9|a^2=ghi,b^2=hi,d^2=i,e^2=gi,f^2=ghi,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, 10124249);