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 | |
4584 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,7)(3,4,8,6)(9,10,14,16)(11,17,19,12)(13,15,21,22)(18,23,24,20)(25,26,31,35)(27,30,37,32)(28,38,36,33)(29,39,40,34);(1,3,5,8)(2,6,7,4)(9,11,14,19)(10,12,16,17)(13,18,21,24)(15,20,22,23)(25,27,31,37)(26,32,35,30)(28,29,36,40)(33,39,38,34);(1,3)(2,4)(5,8)(6,7)(13,21)(15,22)(18,24)(20,23)(25,28,31,36)(26,33,35,38)(27,29,37,40)(30,34,32,39);(1,4)(2,3)(5,6)(7,8)(9,12)(10,11)(13,20)(14,17)(15,18)(16,19)(21,23)(22,24)(25,27)(26,30)(28,29)(31,37)(32,35)(33,34)(36,40)(38,39);(9,10)(11,17)(12,19)(13,15)(14,16)(18,23)(20,24)(21,22)(25,29)(26,34)(27,36)(28,37)(30,38)(31,40)(32,33)(35,39);(9,13)(10,15)(11,18)(12,20)(14,21)(16,22)(17,23)(19,24)(25,30)(26,27)(28,39)(29,38)(31,32)(33,40)(34,36)(35,37)Polycyclic presentation:
2^9|a^2=g,b^2=g,c^2=h,b^a=bg,c^a=ch,d^a=di,e^a=egi,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,f^c=fgi,e^d=egi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8839890);