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 | |
4473 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,3)(4,7,8,6)(9,10,14,15)(11,17,18,16)(12,21,22,13)(19,20,24,23);(2,3)(6,7)(10,15)(11,18)(12,22)(20,23)(25,26)(27,30)(28,32)(29,37)(31,34)(33,39)(35,40)(36,38);(1,3,5,2)(4,7,8,6)(10,15)(13,21)(16,17)(20,23)(25,27,31,28)(26,32,34,30)(29,35,39,36)(33,40,37,38);(1,4)(2,6)(3,7)(5,8)(9,11,14,18)(10,16,15,17)(12,19,22,24)(13,20,21,23)(25,28,31,27)(26,32,34,30)(29,36,39,35)(33,40,37,38);(9,12)(10,13)(11,19)(14,22)(15,21)(16,20)(17,23)(18,24)(25,29)(26,33)(27,35)(28,36)(30,38)(31,39)(32,40)(34,37);(9,13)(10,12)(11,20)(14,21)(15,22)(16,19)(17,24)(18,23)(25,30,31,32)(26,27,34,28)(29,38,39,40)(33,35,37,36)Polycyclic presentation:
2^9|a^2=g,c^2=i,d^2=gi,f^2=ghi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,f^c=fgi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8154842);