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 | #2670 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4080 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10,14,11)(12,15,19,17)(13,18,21,16)(20,23,24,22)(25,26)(27,35)(28,33)(29,34)(30,31)(32,38)(36,40)(37,39);(1,2)(3,5)(4,6)(7,8)(9,10,14,11)(12,17,19,15)(13,16,21,18)(20,23,24,22)(26,31)(32,35)(33,34)(39,40);(1,2,4,6)(3,7,8,5)(10,11)(12,19)(13,21)(22,23)(25,26)(27,32)(28,33)(29,34)(30,31)(35,38)(36,39)(37,40);(3,8)(5,7)(9,11,14,10)(12,15,19,17)(13,18,21,16)(20,22,24,23)(25,27)(26,32)(28,36)(29,37)(30,38)(31,35)(33,39)(34,40);(1,3)(2,5)(4,8)(6,7)(9,12)(10,15)(11,17)(13,20)(14,19)(16,22)(18,23)(21,24)(25,28,30,29)(26,33,31,34)(27,36,38,37)(32,39,35,40);(1,2)(3,5)(4,6)(7,8)(9,13,14,21)(10,16,11,18)(12,20,19,24)(15,22,17,23)(25,29,30,28)(26,34,31,33)(27,37,38,36)(32,40,35,39)Polycyclic presentation:
2^9|a^2=h,b^2=h,c^2=ghi,d^2=h,e^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^c=egi,f^c=fgi,e^d=egi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9399466);