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 | |
4476 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)(25,26)(27,32)(28,29)(30,38)(31,34)(33,40)(35,36)(37,39);(1,3,5,8)(2,6,7,4)(9,10)(11,17)(12,20)(13,14)(15,19)(16,21)(18,23)(22,24)(25,27,31,29)(26,32,34,28)(30,35,40,37)(33,39,38,36);(2,7)(4,6)(10,14)(12,21)(15,17)(18,22)(25,28,31,32)(26,27,34,29)(30,39,40,36)(33,37,38,35);(2,7)(3,8)(9,11)(10,15)(12,22)(13,19)(14,17)(16,24)(18,21)(20,23)(25,26)(27,28)(29,32)(30,33)(31,34)(35,36)(37,39)(38,40);(1,2)(3,6)(4,8)(5,7)(9,12)(10,16)(11,18)(13,21)(14,20)(15,23)(17,24)(19,22)(25,29,31,27)(26,28,34,32)(30,37,40,35)(33,36,38,39);(1,4,5,6)(2,8,7,3)(9,10)(11,15)(12,16)(13,14)(17,19)(18,23)(20,21)(22,24)(25,30,31,40)(26,33,34,38)(27,35,29,37)(28,36,32,39)Polycyclic presentation:
2^9|b^2=h,c^2=gh,e^2=gh,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,e^c=egi,f^c=fgi,e^d=egi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8948146);