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 | |
4577 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,6)(7,8)(9,10)(11,14)(12,16)(13,17)(15,18)(19,22)(20,21)(23,24)(25,26,31,33)(27,32,37,30)(28,34,36,40)(29,35,39,38);(1,2,5,6)(3,7,4,8)(9,11)(10,14)(12,19)(13,18)(15,17)(16,22)(20,23)(21,24)(25,27,31,37)(26,32,33,30)(28,29,36,39)(34,35,40,38);(1,3,5,4)(2,7,6,8)(11,18)(14,15)(19,24)(22,23)(26,33)(30,32)(34,40)(35,38);(1,3,5,4)(2,7,6,8)(9,10)(11,14)(12,20)(13,17)(15,18)(16,21)(19,23)(22,24)(25,28)(26,34)(27,29)(30,38)(31,36)(32,35)(33,40)(37,39);(1,3,5,4)(2,7,6,8)(9,11)(10,15)(12,19)(13,18)(14,17)(16,23)(20,22)(21,24)(25,29)(26,35)(27,36)(28,37)(30,34)(31,39)(32,40)(33,38);(1,4,5,3)(2,8,6,7)(9,12)(10,16)(11,19)(13,21)(14,22)(15,23)(17,20)(18,24)(25,30)(26,27)(28,38)(29,34)(31,32)(33,37)(35,36)(39,40)Polycyclic presentation:
2^9|a^2=h,b^2=gh,c^2=g,d^2=g,e^2=g,f^2=g,b^a=bg,c^a=ch,e^a=ei,c^b=ci,e^c=ei,f^c=fh,e^d=ehi,f^d=fi95 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7976954);