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 | |
4421 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,4,6)(3,5,7,8)(25,26)(27,35)(28,29)(30,32)(31,33)(34,40)(36,37)(38,39);(1,2)(3,5)(4,6)(7,8)(9,10)(11,15)(12,19)(13,14)(16,21)(17,24)(18,22)(20,23)(25,27,31,30)(26,32,33,35)(28,36,34,38)(29,39,40,37);(1,3,4,7)(2,5,6,8)(10,14)(15,22)(16,19)(23,24)(26,33)(27,30)(28,34)(37,39);(9,11,13,18)(10,15,14,22)(12,20,21,17)(16,24,19,23)(25,28)(26,29)(27,36)(30,38)(31,34)(32,39)(33,40)(35,37);(9,12)(10,16)(11,17)(13,21)(14,19)(15,23)(18,20)(22,24)(25,29)(26,34)(27,37)(28,33)(30,39)(31,40)(32,36)(35,38);(9,10)(11,15)(12,16)(13,14)(17,23)(18,22)(19,21)(20,24)(25,30,31,27)(26,32,33,35)(28,38,34,36)(29,39,40,37)Polycyclic presentation:
2^9|a^2=gh,b^2=h,c^2=gh,d^2=hi,f^2=h,b^a=bg,c^a=ch,e^a=eh,c^b=ci,e^b=ei,d^c=dh,f^c=fi,e^d=ei98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7698198);