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 | |
4480 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)(3,4)(5,7)(6,8)(9,10)(11,18)(12,16)(13,21)(14,15)(17,23)(19,20)(22,24)(25,26,30,34)(27,35,36,31)(28,32,38,40)(29,33,37,39);(1,3)(2,6)(4,7)(5,8)(10,15)(12,18)(17,21)(20,22)(27,36)(29,37)(31,35)(33,39);(10,15)(11,16)(17,21)(19,24)(25,27,30,36)(26,31,34,35)(28,37,38,29)(32,33,40,39);(1,2,5,7)(3,6,8,4)(9,11)(10,12)(13,19)(14,16)(15,18)(17,20)(21,22)(23,24)(25,28,30,38)(26,32,34,40)(27,29,36,37)(31,39,35,33);(1,2)(3,6)(4,8)(5,7)(9,12)(10,16)(11,15)(13,22)(14,18)(17,19)(20,23)(21,24)(25,29)(26,33)(27,28)(30,37)(31,40)(32,35)(34,39)(36,38);(1,4,5,6)(2,3,7,8)(9,13,14,23)(10,17,15,21)(11,19,16,24)(12,20,18,22)(25,26)(27,31)(28,32)(29,39)(30,34)(33,37)(35,36)(38,40)Polycyclic presentation:
2^9|a^2=i,c^2=i,d^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^d=egi,f^e=fgi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10013779);