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 |
| prime radical | #56345 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4232 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,6)(3,4,7,8)(11,17)(15,22)(18,20)(23,24)(25,26,31,36)(27,34,29,32)(28,38,37,33)(30,35,39,40);(2,6)(3,7)(9,10)(11,15)(12,19)(13,14)(16,21)(17,22)(18,24)(20,23)(25,27,31,29)(26,32,36,34)(28,39,37,30)(33,40,38,35);(1,3)(2,4)(5,7)(6,8)(10,14)(11,17)(16,19)(18,20)(25,28)(26,33)(27,30)(29,39)(31,37)(32,35)(34,40)(36,38);(1,4)(2,3)(5,8)(6,7)(9,11,13,17)(10,15,14,22)(12,20,21,18)(16,24,19,23)(25,29,31,27)(26,34,36,32)(28,39,37,30)(33,40,38,35);(9,12)(10,16)(11,18)(13,21)(14,19)(15,23)(17,20)(22,24)(25,30,31,39)(26,35,36,40)(27,37,29,28)(32,38,34,33);(9,10)(11,15)(12,16)(13,14)(17,22)(18,23)(19,21)(20,24)(25,26)(27,32)(28,33)(29,34)(30,35)(31,36)(37,38)(39,40)Polycyclic presentation:
2^9|a^2=g,b^2=h,d^2=ghi,e^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dgi,f^c=fgi,e^d=egi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8837472);