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 | |
4393 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,7)(3,8,4,6)(9,10)(11,12)(13,21)(14,15)(16,18)(17,23)(19,22)(20,24)(25,26)(27,31)(28,29)(30,32)(33,37)(34,35)(36,38)(39,40);(3,4)(6,8)(10,15)(12,16)(17,21)(20,22)(25,27,30,36)(26,31,32,38)(28,34,33,39)(29,35,37,40);(1,3,5,4)(2,6,7,8)(10,15)(11,18)(17,21)(19,24)(26,32)(28,33)(31,38)(34,39);(3,4)(6,8)(9,11,14,18)(10,16,15,12)(13,19,23,24)(17,22,21,20)(25,28,30,33)(26,29,32,37)(27,34,36,39)(31,35,38,40);(1,4,5,3)(2,6,7,8)(9,12,14,16)(10,11,15,18)(13,22,23,20)(17,24,21,19)(25,27)(26,31)(28,34)(29,35)(30,36)(32,38)(33,39)(37,40);(1,2)(3,6)(4,8)(5,7)(9,13,14,23)(10,17,15,21)(11,19,18,24)(12,20,16,22)(25,29)(26,33)(27,35)(28,32)(30,37)(31,39)(34,38)(36,40)Polycyclic presentation:
2^9|a^2=i,b^2=ghi,c^2=i,d^2=hi,e^2=gi,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, 10090267);