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 | |
4639 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2)(3,5)(4,7)(6,8)(9,10)(11,19)(12,21)(13,20)(14,18)(15,16)(17,23)(22,24)(25,26)(27,32)(28,29)(30,31)(33,38)(34,37)(35,39)(36,40);(2,3)(6,7)(9,11,15,20)(10,13,16,19)(12,22,23,14)(17,24,21,18)(25,26)(27,34)(28,29)(30,31)(32,37)(33,38)(35,36)(39,40);(1,2,5,3)(4,6,8,7)(10,16)(13,19)(17,21)(18,24)(26,31)(28,38)(32,34)(35,40);(1,3,5,2)(4,7,8,6)(9,12,15,23)(10,17,16,21)(11,14,20,22)(13,18,19,24)(25,27)(26,32)(28,35)(29,39)(30,37)(31,34)(33,36)(38,40);(1,4)(2,6)(3,7)(5,8)(9,13)(10,11)(12,18)(14,17)(15,19)(16,20)(21,22)(23,24)(25,28)(26,33)(27,35)(29,31)(30,38)(32,36)(34,39)(37,40);(9,14)(10,18)(11,12)(13,17)(15,22)(16,24)(19,21)(20,23)(25,29,30,33)(26,28,31,38)(27,36,37,39)(32,40,34,35)Polycyclic presentation:
2^9|b^2=hi,c^2=ghi,d^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,f^d=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8514655);