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 | |
4734 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)(11,16)(12,15)(13,17)(14,19)(18,24)(20,23)(21,22)(25,26)(27,31)(28,29)(30,33)(32,36)(34,35)(37,40)(38,39);(3,7)(5,8)(9,11)(10,14)(12,18)(13,19)(15,22)(16,17)(20,21)(23,24)(25,27)(26,31)(28,34)(29,35)(30,36)(32,33)(37,38)(39,40);(1,2)(3,5)(4,6)(7,8)(12,20)(15,23)(18,21)(22,24)(25,27,30,36)(26,31,33,32)(28,34,38,37)(29,35,39,40);(9,12,13,20)(10,15,17,23)(11,18,19,21)(14,22,16,24)(25,27)(26,32)(28,37)(29,35)(30,36)(31,33)(34,38)(39,40);(1,3)(2,5)(4,7)(6,8)(9,11)(10,16)(12,21)(13,19)(14,17)(15,22)(18,20)(23,24)(25,28,30,38)(26,29,33,39)(27,34,36,37)(31,35,32,40);(1,2)(3,5)(4,6)(7,8)(9,10)(11,14)(12,15)(13,17)(16,19)(18,22)(20,23)(21,24)(25,29)(26,28)(27,35)(30,39)(31,34)(32,37)(33,38)(36,40)Polycyclic presentation:
2^9|a^2=h,c^2=i,d^2=g,e^2=i,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^d=egi,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9113862);