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 | |
4336 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,3)(4,7,8,6)(11,17)(15,22)(18,20)(23,24)(25,26)(27,32)(28,30)(29,35)(31,33)(34,36)(37,38)(39,40);(2,3)(6,7)(9,10)(11,15)(12,19)(13,14)(16,21)(17,22)(18,24)(20,23)(25,27,31,37)(26,32,33,38)(28,35,36,40)(29,34,39,30);(1,3,5,2)(4,7,8,6)(10,14)(15,22)(16,19)(23,24)(26,33)(27,37)(30,34)(35,40);(1,4,5,8)(2,6,3,7)(9,11)(10,15)(12,20)(13,17)(14,22)(16,24)(18,21)(19,23)(25,28,31,36)(26,34,33,30)(27,35,37,40)(29,32,39,38);(9,12)(10,16)(11,18)(13,21)(14,19)(15,23)(17,20)(22,24)(25,29)(26,35)(27,34)(28,38)(30,37)(31,39)(32,36)(33,40);(9,10)(11,15)(12,16)(13,14)(17,22)(18,23)(19,21)(20,24)(25,30,31,34)(26,36,33,28)(27,29,37,39)(32,40,38,35)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=g,d^2=gh,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,f^c=fgi,e^d=egi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8833260);