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 | |
4366 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,4)(3,7,8,6)(9,10,14,15)(11,16,17,12)(13,18,22,23)(19,24,21,20)(25,26,31,35)(27,32,30,34)(28,29,33,40)(36,37,38,39);(1,3,5,8)(2,6,4,7)(9,11)(10,12)(13,21)(14,17)(15,16)(18,24)(19,22)(20,23)(25,27)(26,32)(28,36)(29,37)(30,31)(33,38)(34,35)(39,40);(1,2,5,4)(3,6,8,7)(10,15)(11,17)(13,22)(20,24)(27,30)(32,34)(36,38)(37,39);(1,4,5,2)(3,7,8,6)(9,11)(10,16)(12,15)(13,19)(14,17)(18,24)(20,23)(21,22)(25,28)(26,29)(27,36)(30,38)(31,33)(32,37)(34,39)(35,40);(9,12,14,16)(10,17,15,11)(13,20,22,24)(18,21,23,19)(25,29)(26,33)(27,37)(28,35)(30,39)(31,40)(32,38)(34,36);(9,13,14,22)(10,18,15,23)(11,19,17,21)(12,20,16,24)(25,30,31,27)(26,34,35,32)(28,38,33,36)(29,39,40,37)Polycyclic presentation:
2^9|a^2=ghi,b^2=gh,c^2=gh,d^2=gh,e^2=h,f^2=i,b^a=bg,c^a=ch,e^a=eh,c^b=ci,f^b=fi,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, 7747096);