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 | |
4295 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,6,7)(3,5,4,8)(9,10)(11,14)(12,15)(13,16)(17,20)(18,22)(19,23)(21,24);(2,7)(3,4)(10,11)(15,17)(16,18)(23,24)(25,26)(27,33)(28,29)(30,34)(31,37)(32,38)(35,39)(36,40);(1,3,6,4)(2,8,7,5)(25,27)(26,31)(28,35)(29,36)(30,37)(32,39)(33,34)(38,40);(1,4,6,3)(2,5,7,8)(9,11,14,10)(12,17,20,15)(13,16,22,18)(19,23,21,24)(25,28,30,38)(26,32,34,29)(27,35,37,40)(31,39,33,36);(1,5)(2,3)(4,7)(6,8)(9,12,14,20)(10,15,11,17)(13,21,22,19)(16,24,18,23)(25,27)(26,33)(28,35)(29,39)(30,37)(31,34)(32,36)(38,40);(9,13,14,22)(10,16,11,18)(12,19,20,21)(15,23,17,24)(25,29,30,32)(26,28,34,38)(27,36,37,39)(31,35,33,40)Polycyclic presentation:
2^9|a^2=h,c^2=h,d^2=ghi,e^2=gh,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^d=fgi,f^e=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8608494);