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 | #35405 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4354 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)(3,8)(4,7)(5,6)(9,10,15,18)(11,16,19,14)(12,13,17,24)(20,21,22,23)(25,26)(27,28)(29,33)(30,39)(31,34)(32,38)(35,37)(36,40);(1,3)(2,4)(5,8)(6,7)(9,11,15,19)(10,16,18,14)(12,22,17,20)(13,23,24,21)(25,27,31,32)(26,28,34,38)(29,35,39,40)(30,36,33,37);(2,5)(4,8)(11,19)(12,17)(13,24)(14,16)(25,28,31,38)(26,32,34,27)(29,36,39,37)(30,40,33,35);(1,2,6,5)(3,8,7,4)(9,12)(10,13)(11,20)(14,23)(15,17)(16,21)(18,24)(19,22)(25,28)(26,27)(29,36)(30,35)(31,38)(32,34)(33,40)(37,39);(1,4,6,8)(2,7,5,3)(9,13)(10,17)(11,21)(12,18)(14,20)(15,24)(16,22)(19,23)(25,29)(26,33)(27,35)(28,37)(30,34)(31,39)(32,40)(36,38);(1,5,6,2)(3,8,7,4)(9,14)(10,11)(12,23)(13,20)(15,16)(17,21)(18,19)(22,24)(25,30,31,33)(26,29,34,39)(27,36,32,37)(28,35,38,40)Polycyclic presentation:
2^9|a^2=ghi,b^2=i,c^2=gh,d^2=g,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,d^c=dgi,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, 8969294);