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 | |
4469 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10)(11,19)(12,17)(13,22)(14,16)(15,18)(20,24)(21,23)(25,26,30,29)(27,31,35,38)(28,36,37,32)(33,40,34,39);(1,2)(3,8)(4,6)(5,7)(9,11,15,14)(10,16,18,19)(12,20,13,21)(17,23,22,24)(25,27)(26,31)(28,34)(29,38)(30,35)(32,40)(33,37)(36,39);(1,3,5,6)(2,4,7,8)(9,12,15,13)(10,17,18,22)(11,20,14,21)(16,23,19,24)(25,28)(26,32)(27,33)(29,36)(30,37)(31,39)(34,35)(38,40);(1,2)(3,4)(5,7)(6,8)(9,13,15,12)(10,17,18,22)(11,21,14,20)(16,23,19,24)(25,29,30,26)(27,31,35,38)(28,36,37,32)(33,39,34,40);(1,2)(3,4)(5,7)(6,8)(9,14,15,11)(10,16,18,19)(12,21,13,20)(17,23,22,24)(25,28)(26,32)(27,34)(29,36)(30,37)(31,40)(33,35)(38,39);(1,4)(2,6)(3,7)(5,8)(9,10)(11,16)(12,17)(13,22)(14,19)(15,18)(20,23)(21,24)(25,27)(26,31)(28,33)(29,38)(30,35)(32,39)(34,37)(36,40)Polycyclic presentation:
2^9|a^2=h,b^2=g,c^2=ghi,d^2=gh,e^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^d=fgi,f^e=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8571975);