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 | |
4513 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)(9,10,15,18)(11,17,12,16)(13,23,24,14)(19,22,21,20)(28,36)(32,37)(35,40)(38,39);(2,3)(6,7)(9,11,15,12)(10,16,18,17)(13,21,24,19)(14,20,23,22)(25,26)(27,31)(28,37)(29,33)(30,34)(32,36)(35,38)(39,40);(1,2,5,3)(4,6,8,7)(25,27)(26,29)(28,35)(30,33)(31,34)(32,38)(36,40)(37,39);(1,3,5,2)(4,7,8,6)(9,12,15,11)(10,17,18,16)(13,21,24,19)(14,20,23,22)(25,27)(26,31)(28,35)(29,34)(30,33)(32,39)(36,40)(37,38);(1,4,5,8)(2,6,3,7)(9,13)(10,14)(11,19)(12,21)(15,24)(16,22)(17,20)(18,23)(25,28)(26,32)(27,35)(29,38)(30,36)(31,39)(33,40)(34,37);(9,14)(10,13)(11,20)(12,22)(15,23)(16,21)(17,19)(18,24)(25,29,30,31)(26,33,34,27)(28,38,36,39)(32,40,37,35)Polycyclic presentation:
2^9|a^2=gh,b^2=gh,c^2=h,d^2=g,e^2=h,f^2=hi,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,e^b=egi,f^d=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8405022);