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 | |
4451 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2)(3,4)(5,7)(6,8)(9,10)(11,13)(12,15)(14,20)(16,18)(17,22)(19,23)(21,24)(25,26,31,35)(27,32,37,30)(28,33,36,40)(29,34,38,39);(1,3)(2,6)(4,7)(5,8)(9,11)(10,16)(12,13)(14,19)(15,18)(17,23)(20,21)(22,24)(25,27)(26,32)(28,38)(29,36)(30,35)(31,37)(33,39)(34,40);(10,12)(11,18)(17,20)(19,24)(25,28,31,36)(26,33,35,40)(27,29,37,38)(30,39,32,34);(2,7)(4,6)(9,12,15,10)(11,13,18,16)(14,20,22,17)(19,21,24,23)(25,27)(26,32)(28,29)(30,35)(31,37)(33,34)(36,38)(39,40);(1,4,5,6)(2,3,7,8)(9,13)(10,11)(12,18)(14,21)(15,16)(17,19)(20,24)(22,23)(25,29)(26,34)(27,36)(28,37)(30,33)(31,38)(32,40)(35,39);(1,3)(2,4)(5,8)(6,7)(9,14)(10,17)(11,19)(12,20)(13,21)(15,22)(16,23)(18,24)(25,30,31,32)(26,27,35,37)(28,39,36,34)(29,40,38,33)Polycyclic presentation:
2^9|a^2=hi,c^2=hi,d^2=h,e^2=gh,f^2=hi,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, 10103088);