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 | |
4638 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,5)(6,8)(9,10)(11,14)(12,16)(13,17)(15,19)(18,22)(20,21)(23,24)(25,26,31,29)(27,35,38,32)(28,33,37,39)(30,34,36,40);(3,7)(6,8)(9,11)(10,14)(12,18)(13,19)(15,17)(16,22)(20,23)(21,24)(25,27)(26,32)(28,36)(29,35)(30,37)(31,38)(33,34)(39,40);(1,2)(3,6)(4,5)(7,8)(9,11,13,19)(10,14,17,15)(12,18,21,24)(16,22,20,23)(25,28,31,37)(26,33,29,39)(27,36,38,30)(32,34,35,40);(1,3,4,7)(2,6,5,8)(9,11)(10,15)(12,18)(13,19)(14,17)(16,23)(20,22)(21,24)(25,28)(26,33)(27,36)(29,39)(30,38)(31,37)(32,34)(35,40);(1,2)(3,6)(4,5)(7,8)(9,12)(10,16)(11,18)(13,21)(14,22)(15,23)(17,20)(19,24)(25,29,31,26)(27,35,38,32)(28,39,37,33)(30,34,36,40);(9,10)(11,14)(12,20)(13,17)(15,19)(16,21)(18,23)(22,24)(25,30,31,36)(26,34,29,40)(27,37,38,28)(32,33,35,39)Polycyclic presentation:
2^9|a^2=g,c^2=gi,d^2=h,e^2=g,f^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, 8646687);