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 | |
4696 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,4)(3,7,8,6)(9,10,14,16)(11,17,21,15)(12,18,23,13)(19,20,22,24)(25,26,31,33)(27,36,37,32)(28,30,35,40)(29,39,38,34);(1,3,5,8)(2,6,4,7)(9,11)(10,15)(12,19)(13,20)(14,21)(16,17)(18,24)(22,23)(25,27)(26,32)(28,29)(30,34)(31,37)(33,36)(35,38)(39,40);(10,16)(13,18)(15,17)(20,24)(26,33)(27,37)(29,38)(30,40);(9,11)(10,17)(12,22)(13,20)(14,21)(15,16)(18,24)(19,23)(25,28)(26,30)(27,38)(29,37)(31,35)(32,39)(33,40)(34,36);(1,2,5,4)(3,6,8,7)(9,12)(10,18)(11,19)(13,16)(14,23)(15,24)(17,20)(21,22)(25,29,31,38)(26,34,33,39)(27,28,37,35)(30,36,40,32);(1,4,5,2)(3,7,8,6)(9,13,14,18)(10,12,16,23)(11,20,21,24)(15,19,17,22)(25,30)(26,35)(27,34)(28,33)(29,36)(31,40)(32,38)(37,39)Polycyclic presentation:
2^9|a^2=g,b^2=gh,e^2=ghi,f^2=gi,b^a=bg,c^a=ch,e^a=ei,c^b=ci,d^b=di,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, 7998921);