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 |
| prime radical | #39396 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4457 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)(9,10,15,17)(11,20,22,16)(12,18,21,14)(13,19,23,24)(25,26)(27,32)(28,30)(29,36)(31,34)(33,39)(35,37)(38,40);(1,2,5,6)(3,7,4,8)(9,11,15,22)(10,16,17,20)(12,13,21,23)(14,19,18,24)(25,27)(26,32)(28,36)(29,30)(31,35)(33,38)(34,37)(39,40);(1,3,5,4)(2,7,6,8)(10,17)(14,18)(16,20)(19,24)(27,35)(29,39)(32,37)(33,36);(1,3,5,4)(2,7,6,8)(9,12)(10,18)(11,13)(14,17)(15,21)(16,24)(19,20)(22,23)(25,28,31,38)(26,30,34,40)(27,33,35,36)(29,32,39,37);(9,13)(10,19)(11,21)(12,22)(14,20)(15,23)(16,18)(17,24)(25,29)(26,33)(27,30)(28,37)(31,39)(32,38)(34,36)(35,40);(1,4,5,3)(2,8,6,7)(9,14,15,18)(10,12,17,21)(11,19,22,24)(13,20,23,16)(25,30)(26,28)(27,29)(31,40)(32,36)(33,37)(34,38)(35,39)Polycyclic presentation:
2^9|a^2=h,b^2=g,c^2=gh,d^2=ghi,f^2=g,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, 7995531);