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 | |
4323 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)(5,7)(6,8)(9,10,14,15)(11,16,21,17)(12,18,23,13)(19,20,22,24);(1,3,5,8)(2,6,7,4)(10,15)(13,18)(16,17)(20,24)(25,26)(27,30)(28,36)(29,32)(31,37)(33,35)(34,39)(38,40);(2,7)(4,6)(9,11,14,21)(10,16,15,17)(12,22,23,19)(13,20,18,24)(25,26,29,32)(27,33,35,30)(28,31,37,36)(34,39,40,38);(1,2,5,7)(3,6,8,4)(9,11)(10,17)(12,19)(13,20)(14,21)(15,16)(18,24)(22,23)(25,27,29,35)(26,30,32,33)(28,34,37,40)(31,38,36,39);(1,4,5,6)(2,3,7,8)(9,12)(10,18)(11,19)(13,15)(14,23)(16,20)(17,24)(21,22)(25,26)(27,30)(28,31)(29,32)(33,35)(34,38)(36,37)(39,40);(1,3)(2,6)(4,7)(5,8)(9,13,14,18)(10,12,15,23)(11,20,21,24)(16,22,17,19)(25,28)(26,31)(27,34)(29,37)(30,38)(32,36)(33,39)(35,40)Polycyclic presentation:
2^9|a^2=gh,b^2=h,c^2=ghi,d^2=hi,e^2=h,f^2=gh,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, 8690424);