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 | |
4234 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10,15,17)(11,14,12,16)(13,18,22,24)(19,21,20,23)(25,26,31,34)(27,35,37,32)(28,38,39,30)(29,33,40,36);(2,3)(6,7)(9,11,15,12)(10,16,17,14)(13,20,22,19)(18,21,24,23)(25,27,31,37)(26,32,34,35)(28,36,39,33)(29,38,40,30);(1,2,5,3)(4,6,8,7)(10,17)(11,12)(18,24)(19,20);(9,12,15,11)(10,14,17,16)(13,20,22,19)(18,21,24,23)(25,28)(26,30)(27,36)(29,32)(31,39)(33,37)(34,38)(35,40);(1,3,5,2)(4,6,8,7)(9,13,15,22)(10,18,17,24)(11,19,12,20)(14,21,16,23)(25,29,31,40)(26,33,34,36)(27,30,37,38)(28,32,39,35);(1,4)(2,6)(3,7)(5,8)(9,14)(10,11)(12,17)(13,21)(15,16)(18,19)(20,24)(22,23)(25,30)(26,28)(27,29)(31,38)(32,36)(33,35)(34,39)(37,40)Polycyclic presentation:
2^9|a^2=g,b^2=g,c^2=hi,d^2=h,e^2=ghi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eghi,d^c=dh,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9756112);