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 | #39101 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4222 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)(5,8)(6,7)(9,10,14,18)(11,13,17,15)(12,16,22,20)(19,23,24,21)(25,26)(27,38)(28,31)(29,39)(30,35)(32,37)(33,36)(34,40);(1,3,5,7)(2,4,8,6)(9,11,14,17)(10,15,18,13)(12,19,22,24)(16,21,20,23)(25,27)(26,30)(28,39)(29,33)(31,40)(32,35)(34,36)(37,38);(1,3)(2,6)(4,8)(5,7)(25,28,32,36)(26,33,37,31)(27,39,35,34)(29,38,40,30);(1,3,5,7)(2,4,8,6)(9,10)(11,15)(12,20)(13,17)(14,18)(16,22)(19,23)(21,24)(25,29,32,40)(26,34,37,39)(27,33,35,31)(28,30,36,38);(1,3)(2,4)(5,7)(6,8)(9,12)(10,16)(11,19)(13,23)(14,22)(15,21)(17,24)(18,20)(25,30)(26,35)(27,37)(28,29)(31,39)(32,38)(33,34)(36,40);(1,4)(2,7)(3,8)(5,6)(9,13,14,15)(10,17,18,11)(12,21,22,23)(16,19,20,24)(25,31,32,33)(26,36,37,28)(27,40,35,29)(30,34,38,39)Polycyclic presentation:
2^9|a^2=ghi,b^2=gh,c^2=hi,d^2=h,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, 10014183);