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 | |
4354 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)(3,6)(4,5)(7,8)(25,26,30,31)(27,33,36,32)(28,29,37,40)(34,35,38,39);(1,3,5,8)(2,6,4,7)(9,10,14,15)(11,16,20,23)(12,13,17,22)(18,24,21,19)(26,31)(28,37)(32,33)(34,38);(1,3)(2,7)(4,6)(5,8)(10,15)(13,22)(16,23)(19,24)(25,27,30,36)(26,32,31,33)(28,34,37,38)(29,39,40,35);(1,3,5,8)(2,6,4,7)(9,11)(10,16)(12,21)(13,19)(14,20)(15,23)(17,18)(22,24)(25,27)(26,32)(28,38)(29,35)(30,36)(31,33)(34,37)(39,40);(1,3)(2,6)(4,7)(5,8)(9,12,14,17)(10,13,15,22)(11,18,20,21)(16,24,23,19)(25,28)(26,29)(27,34)(30,37)(31,40)(32,39)(33,35)(36,38);(1,4,5,2)(3,7,8,6)(9,13)(10,17)(11,19)(12,15)(14,22)(16,18)(20,24)(21,23)(25,29)(26,28)(27,35)(30,40)(31,37)(32,38)(33,34)(36,39)Polycyclic presentation:
2^9|a^2=g,b^2=i,c^2=g,d^2=gh,e^2=ghi,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,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, 9360258);