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 | |
4420 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(3,7)(5,8)(9,10)(11,15)(12,19)(13,21)(14,17)(16,22)(18,20)(23,24)(25,26,32,31)(27,35,30,33)(28,38,34,29)(36,37,39,40);(1,2)(3,5)(4,6)(7,8)(9,11,14,19)(10,15,17,12)(13,18,22,24)(16,23,21,20)(25,27,32,30)(26,33,31,35)(28,39,34,36)(29,40,38,37);(1,2,4,6)(3,5,7,8)(9,10,14,17)(11,12,19,15)(13,16,22,21)(18,20,24,23)(25,28)(26,29)(27,36)(30,39)(31,38)(32,34)(33,37)(35,40);(1,2)(3,5)(4,6)(7,8)(9,10)(11,12)(13,16)(14,17)(15,19)(18,20)(21,22)(23,24)(25,29,32,38)(26,34,31,28)(27,37,30,40)(33,39,35,36);(1,2)(3,5)(4,6)(7,8)(9,12)(10,11)(13,20)(14,15)(16,18)(17,19)(21,24)(22,23)(25,30,32,27)(26,35,31,33)(28,39,34,36)(29,40,38,37);(1,3)(2,5)(4,7)(6,8)(9,13,14,22)(10,16,17,21)(11,18,19,24)(12,20,15,23)(25,31,32,26)(27,35,30,33)(28,38,34,29)(36,40,39,37)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=gi,d^2=g,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi,e^c=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8034304);