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 | #59738 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4416 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,3)(6,7)(9,10,15,19)(11,20,14,16)(12,22,23,17)(13,18,24,21)(25,26)(27,31)(28,36)(29,33)(30,35)(32,37)(34,38)(39,40);(1,2,5,3)(4,6,8,7)(9,11,15,14)(10,16,19,20)(12,18,23,21)(13,17,24,22)(25,26)(27,31)(28,32)(29,33)(30,35)(34,39)(36,37)(38,40);(1,2,5,3)(4,6,8,7)(25,27)(26,30)(28,34)(29,35)(31,33)(32,38)(36,39)(37,40);(1,2,5,3)(4,7,8,6)(9,12,15,23)(10,17,19,22)(11,21,14,18)(13,20,24,16)(25,27)(26,31)(28,34)(29,35)(30,33)(32,39)(36,38)(37,40);(1,3,5,2)(4,6,8,7)(9,13)(10,18)(11,17)(12,20)(14,22)(15,24)(16,23)(19,21)(25,28,29,37)(26,32,33,36)(27,34,35,40)(30,38,31,39);(1,4,5,8)(2,6,3,7)(9,14,15,11)(10,16,19,20)(12,21,23,18)(13,17,24,22)(25,26)(27,30)(28,36)(29,33)(31,35)(32,37)(34,39)(38,40)Polycyclic presentation:
2^9|a^2=ghi,b^2=g,c^2=hi,d^2=g,e^2=h,f^2=g,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, 8603345);