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 | |
4325 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,6)(3,4,8,7)(9,10,15,18)(11,16,12,14)(13,17,22,24)(19,23,20,21)(25,26,31,28)(27,34,30,32)(29,33,38,36)(35,40,37,39);(2,6)(4,7)(11,12)(14,16)(19,20)(21,23)(25,26,31,28)(27,32,30,34)(29,33,38,36)(35,39,37,40);(1,3,5,8)(2,7,6,4)(9,11,15,12)(10,16,18,14)(13,20,22,19)(17,21,24,23)(25,27,31,30)(26,32,28,34)(29,35,38,37)(33,39,36,40);(1,4)(2,3)(5,7)(6,8)(9,12,15,11)(10,14,18,16)(13,19,22,20)(17,23,24,21)(25,28,31,26)(27,34,30,32)(29,33,38,36)(35,39,37,40);(1,3)(2,4)(5,8)(6,7)(9,13)(10,17)(11,19)(12,20)(14,21)(15,22)(16,23)(18,24)(25,29,31,38)(26,33,28,36)(27,35,30,37)(32,39,34,40);(1,4)(2,3)(5,7)(6,8)(9,14)(10,11)(12,18)(13,21)(15,16)(17,19)(20,24)(22,23)(25,30,31,27)(26,34,28,32)(29,37,38,35)(33,40,36,39)Polycyclic presentation:
2^9|a^2=hi,b^2=gh,c^2=hi,d^2=ghi,e^2=gh,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,f^b=fgi,e^c=egi,e^d=eghi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10273360);