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 | |
4408 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10,14,15)(11,17,18,12)(13,21,22,16)(19,20,24,23)(25,26,31,33)(27,32,30,40)(28,29,34,39)(35,38,37,36);(1,2)(3,8)(4,6)(5,7)(10,15)(12,17)(16,21)(20,23)(25,27,31,30)(26,32,33,40)(28,35,34,37)(29,38,39,36);(1,3)(2,4)(5,6)(7,8)(10,15)(11,18)(13,22)(20,23)(26,33)(27,30)(28,34)(36,38);(3,6)(4,8)(9,11,14,18)(10,12,15,17)(13,19,22,24)(16,20,21,23)(25,28)(26,29)(27,35)(30,37)(31,34)(32,38)(33,39)(36,40);(1,4,5,8)(2,6,7,3)(9,12)(10,11)(13,20)(14,17)(15,18)(16,19)(21,24)(22,23)(25,29)(26,34)(27,36)(28,33)(30,38)(31,39)(32,35)(37,40);(1,2)(3,8)(4,6)(5,7)(9,13)(10,16)(11,19)(12,20)(14,22)(15,21)(17,23)(18,24)(25,30,31,27)(26,32,33,40)(28,37,34,35)(29,38,39,36)Polycyclic presentation:
2^9|a^2=h,b^2=gh,d^2=g,e^2=ghi,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,f^c=fgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8821645);