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 | |
4536 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10,14,15)(11,17,19,12)(13,21,23,16)(18,20,22,24)(25,26,30,35)(27,31,38,32)(28,33,34,29)(36,40,39,37);(1,2)(3,8)(4,5)(6,7)(9,11,14,19)(10,12,15,17)(13,22,23,18)(16,24,21,20)(25,27)(26,31)(28,39)(29,40)(30,38)(32,35)(33,37)(34,36);(2,7)(3,4)(10,15)(12,17)(16,21)(20,24)(25,27,30,38)(26,32,35,31)(28,39,34,36)(29,37,33,40);(1,3,6,4)(2,8,7,5)(9,10)(11,12)(13,16)(14,15)(17,19)(18,20)(21,23)(22,24)(25,27)(26,31)(28,36)(29,37)(30,38)(32,35)(33,40)(34,39);(1,4,6,3)(2,5,7,8)(9,12)(10,11)(13,20)(14,17)(15,19)(16,18)(21,22)(23,24)(25,28,30,34)(26,33,35,29)(27,36,38,39)(31,40,32,37);(1,5)(2,3)(4,7)(6,8)(9,13,14,23)(10,16,15,21)(11,18,19,22)(12,20,17,24)(25,29,30,33)(26,34,35,28)(27,37,38,40)(31,39,32,36)Polycyclic presentation:
2^9|a^2=h,b^2=g,c^2=gh,d^2=ghi,e^2=i,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^c=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8706667);