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 | |
4506 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,6)(7,8)(9,10,14,15)(11,17,19,12)(13,16,21,22)(18,23,24,20)(25,26,31,30)(27,35,36,32)(28,33,37,38)(29,39,40,34);(1,2)(3,7)(4,8)(5,6)(9,11)(10,12)(13,18)(14,19)(15,17)(16,20)(21,24)(22,23)(25,27,31,36)(26,32,30,35)(28,29,37,40)(33,34,38,39);(10,15)(12,17)(16,22)(20,23)(26,30)(27,36)(28,37)(34,39);(9,12)(10,11)(13,20)(14,17)(15,19)(16,18)(21,23)(22,24)(25,28)(26,33)(27,29)(30,38)(31,37)(32,34)(35,39)(36,40);(1,3,5,4)(2,7,6,8)(9,11)(10,12)(13,18)(14,19)(15,17)(16,20)(21,24)(22,23)(25,29)(26,34)(27,37)(28,36)(30,39)(31,40)(32,38)(33,35);(1,4,5,3)(2,8,6,7)(9,13,14,21)(10,16,15,22)(11,18,19,24)(12,20,17,23)(25,30,31,26)(27,35,36,32)(28,38,37,33)(29,39,40,34)Polycyclic presentation:
2^9|a^2=h,b^2=i,e^2=gh,f^2=g,b^a=bg,c^a=ch,e^a=eh,c^b=ci,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, 7673797);