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 | |
4442 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)(3,7)(4,6)(5,8)(9,10)(11,18)(12,22)(13,17)(14,15)(16,19)(20,24)(21,23)(25,26,31,34)(27,29,28,32)(30,33,38,40)(35,37,36,39);(1,2,4,6)(3,5,8,7)(10,15)(11,19)(12,17)(21,24)(25,27,31,28)(26,32,34,29)(30,35,38,36)(33,39,40,37);(1,2)(3,5)(4,6)(7,8)(10,15)(13,22)(16,18)(21,24)(30,38)(33,40)(35,36)(37,39);(1,3,4,8)(2,5,6,7)(9,11)(10,16)(12,23)(13,21)(14,19)(15,18)(17,20)(22,24)(25,28,31,27)(26,29,34,32)(30,36,38,35)(33,37,40,39);(9,12,14,17)(10,13,15,22)(11,20,19,23)(16,24,18,21)(25,29)(26,28)(27,34)(30,37)(31,32)(33,36)(35,40)(38,39);(9,13)(10,17)(11,21)(12,15)(14,22)(16,20)(18,23)(19,24)(25,30,31,38)(26,33,34,40)(27,35,28,36)(29,37,32,39)Polycyclic presentation:
2^9|a^2=ghi,b^2=gh,d^2=gh,e^2=h,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=fgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8281398);