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 | |
4448 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)(3,6,7,4)(9,10,15,17)(11,16,14,24)(12,18,21,13)(19,20,22,23)(25,26)(27,37)(28,34)(29,30)(31,35)(32,36)(33,39)(38,40);(1,3,5,7)(2,6,8,4)(9,11,15,14)(10,16,17,24)(12,19,21,22)(13,23,18,20)(25,27)(26,33)(28,31)(29,38)(30,35)(32,39)(34,40)(36,37);(1,3)(2,4)(5,7)(6,8)(10,17)(12,21)(16,24)(19,22)(25,28,32,29)(26,34,36,30)(27,31,39,38)(33,40,37,35);(1,3,5,7)(2,6,8,4)(9,12)(10,18)(11,19)(13,17)(14,22)(15,21)(16,20)(23,24)(25,29,32,28)(26,34,36,30)(27,38,39,31)(33,40,37,35);(1,3)(2,6)(4,8)(5,7)(9,13,15,18)(10,12,17,21)(11,20,14,23)(16,22,24,19)(25,30,32,34)(26,29,36,28)(27,35,39,40)(31,33,38,37);(1,4,5,6)(2,7,8,3)(9,14,15,11)(10,16,17,24)(12,22,21,19)(13,23,18,20)(25,31,32,38)(26,35,36,40)(27,28,39,29)(30,37,34,33)Polycyclic presentation:
2^9|a^2=h,b^2=h,c^2=g,d^2=gi,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=dh,e^d=egi,f^e=fgi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10075596);