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 | |
4600 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(4,8)(6,7)(9,10,14,15)(11,18,16,12)(13,21,22,17)(19,20,23,24)(25,26)(27,34)(28,32)(29,31)(30,37)(33,38)(35,39)(36,40);(1,2,5,3)(4,6,8,7)(10,15)(11,16)(17,21)(19,23)(27,28)(32,34)(35,36)(39,40);(2,3)(6,7)(10,15)(12,18)(17,21)(20,24)(25,27,31,28)(26,32,29,34)(30,35,38,36)(33,39,37,40);(9,11,14,16)(10,12,15,18)(13,19,22,23)(17,20,21,24)(25,28,31,27)(26,32,29,34)(30,36,38,35)(33,39,37,40);(1,3,5,2)(4,7,8,6)(9,12)(10,16)(11,15)(13,20)(14,18)(17,23)(19,21)(22,24)(25,29,31,26)(27,32,28,34)(30,37,38,33)(35,39,36,40);(1,4)(2,6)(3,7)(5,8)(9,13,14,22)(10,17,15,21)(11,19,16,23)(12,20,18,24)(25,30)(26,33)(27,35)(28,36)(29,37)(31,38)(32,39)(34,40)Polycyclic presentation:
2^9|a^2=g,b^2=ghi,c^2=gh,d^2=h,e^2=i,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^c=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8027273);