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 | #59576 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4286 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,8)(3,6,4,7)(9,10)(11,20)(12,13)(14,19)(15,17)(16,22)(18,23)(21,24);(1,2)(3,6)(4,7)(5,8)(9,11,15,22)(10,16,17,20)(12,19,18,24)(13,21,23,14)(25,26)(27,32)(28,33)(29,35)(30,31)(34,39)(36,38)(37,40);(1,3,5,4)(2,6,8,7)(10,17)(12,18)(16,20)(19,24)(26,31)(29,39)(32,40)(33,36);(9,12,15,18)(10,13,17,23)(11,19,22,24)(14,16,21,20)(25,27)(26,32)(28,34)(29,36)(30,37)(31,40)(33,39)(35,38);(1,3,5,4)(2,6,8,7)(9,13)(10,18)(11,21)(12,17)(14,22)(15,23)(16,24)(19,20)(25,28)(26,33)(27,35)(29,32)(30,38)(31,36)(34,37)(39,40);(1,4,5,3)(2,7,8,6)(9,14,15,21)(10,19,17,24)(11,13,22,23)(12,16,18,20)(25,29)(26,34)(27,36)(28,32)(30,39)(31,35)(33,37)(38,40)Polycyclic presentation:
2^9|a^2=gh,b^2=h,c^2=gh,d^2=h,e^2=gh,f^2=g,b^a=bg,c^a=ch,e^a=eh,c^b=ci,f^b=fi,d^c=dh,f^c=fi,e^d=ei98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7721006);