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 |
| prime radical | #2671 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4083 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)(9,10)(11,13)(12,21)(14,23)(15,16)(17,22)(18,20)(19,24)(25,26)(27,37)(28,31)(29,34)(30,35)(32,36)(33,40)(38,39);(2,3)(6,7)(9,11,15,19)(10,13,16,24)(12,20,22,14)(17,23,21,18)(25,27)(26,33)(28,30)(29,38)(31,39)(32,40)(34,35)(36,37);(1,3,5,2)(4,7,8,6)(10,16)(11,19)(12,22)(18,23)(25,28,32,29)(26,31,36,34)(27,30,40,38)(33,39,37,35);(1,4,5,8)(2,6,3,7)(9,12)(10,17)(11,20)(13,23)(14,19)(15,22)(16,21)(18,24)(25,29,32,28)(26,34,36,31)(27,38,40,30)(33,35,37,39);(9,13)(10,11)(12,23)(14,21)(15,24)(16,19)(17,20)(18,22)(25,30,32,38)(26,35,36,39)(27,28,40,29)(31,33,34,37);(9,14,15,20)(10,18,16,23)(11,12,19,22)(13,17,24,21)(25,31,32,34)(26,28,36,29)(27,39,40,35)(30,37,38,33)Polycyclic presentation:
2^9|a^2=hi,b^2=h,c^2=g,d^2=g,e^2=ghi,f^2=gi,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,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, 9626369);