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 | |
4581 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,4,6)(3,5,7,8)(9,10)(11,15)(12,18)(13,17)(14,16)(19,22)(20,21)(23,24)(25,26,31,35)(27,36,38,32)(28,34,39,33)(29,40,37,30);(1,3,4,7)(2,5,6,8)(9,11,14,18)(10,15,16,12)(13,19,23,21)(17,22,24,20)(25,27)(26,32)(28,29)(30,34)(31,38)(33,40)(35,36)(37,39);(1,2)(3,5)(4,6)(7,8)(10,16)(11,18)(17,24)(19,21)(25,28,31,39)(26,33,35,34)(27,29,38,37)(30,32,40,36);(1,3)(2,5)(4,7)(6,8)(9,11)(10,12)(13,21)(14,18)(15,16)(17,22)(19,23)(20,24)(25,28)(26,34)(27,37)(29,38)(30,36)(31,39)(32,40)(33,35);(1,2)(3,5)(4,6)(7,8)(9,12)(10,11)(13,22)(14,15)(16,18)(17,21)(19,24)(20,23)(25,29,31,37)(26,30,35,40)(27,28,38,39)(32,34,36,33);(9,13)(10,17)(11,19)(12,20)(14,23)(15,22)(16,24)(18,21)(25,30)(26,29)(27,34)(28,32)(31,40)(33,38)(35,37)(36,39)Polycyclic presentation:
2^9|a^2=hi,b^2=gh,c^2=g,e^2=g,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8637909);