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 | |
4592 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2)(3,4)(5,6)(7,8)(9,10)(11,18)(12,16)(13,21)(14,15)(17,22)(19,20)(23,24)(25,26,31,33)(27,34,28,32)(29,30,39,40)(35,38,37,36);(2,6)(3,7)(10,15)(11,16)(13,22)(20,24);(1,3,5,7)(2,4,6,8)(10,15)(12,18)(17,21)(20,24)(25,27,31,28)(26,32,33,34)(29,35,39,37)(30,36,40,38);(1,4)(2,7)(3,6)(5,8)(9,11,14,16)(10,12,15,18)(13,19,22,23)(17,20,21,24)(25,28,31,27)(26,32,33,34)(29,35,39,37)(30,38,40,36);(9,12)(10,16)(11,15)(13,20)(14,18)(17,23)(19,21)(22,24)(25,29)(26,30)(27,35)(28,37)(31,39)(32,36)(33,40)(34,38);(9,13)(10,17)(11,19)(12,20)(14,22)(15,21)(16,23)(18,24)(25,30)(26,29)(27,36)(28,38)(31,40)(32,35)(33,39)(34,37)Polycyclic presentation:
2^9|a^2=ghi,c^2=gh,d^2=h,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, 10232420);