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 | |
4403 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,8)(3,6,4,7)(9,10)(11,21)(12,17)(13,18)(14,19)(15,20)(16,22)(23,24)(25,26)(27,32)(28,34)(29,38)(30,35)(31,33)(36,39)(37,40);(1,2)(3,6)(4,7)(5,8)(9,11)(10,16)(12,23)(13,24)(14,18)(15,22)(17,19)(20,21)(25,27,31,36)(26,32,33,39)(28,35,38,40)(29,37,34,30);(1,3,5,4)(2,6,8,7)(11,22)(14,23)(16,21)(19,24)(26,33)(27,36)(28,38)(30,37);(9,12)(10,17)(11,23)(13,20)(14,22)(15,18)(16,19)(21,24)(25,28,31,38)(26,34,33,29)(27,35,36,40)(30,39,37,32);(1,4,5,3)(2,7,8,6)(9,13,15,17)(10,18,20,12)(11,24,22,19)(14,21,23,16)(25,29,31,34)(26,28,33,38)(27,37,36,30)(32,35,39,40);(9,14,15,23)(10,19,20,24)(11,12,22,18)(13,21,17,16)(25,30,31,37)(26,35,33,40)(27,34,36,29)(28,39,38,32)Polycyclic presentation:
2^9|a^2=ghi,b^2=h,c^2=ghi,d^2=h,e^2=gh,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, 7751758);