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 | |
4406 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)(3,5)(4,8)(6,7)(9,10,14,15)(11,13,17,23)(12,19,20,16)(18,24,21,22)(25,26)(27,31)(28,29)(30,32)(33,36)(34,35)(37,38)(39,40);(1,3)(2,5)(4,6)(7,8)(10,15)(11,17)(12,20)(22,24)(25,27,30,36)(26,31,32,33)(28,37,35,39)(29,38,34,40);(1,2)(3,8)(4,5)(6,7)(11,17)(13,23)(18,21)(22,24)(26,32)(27,36)(29,34)(37,39);(2,7)(3,4)(9,11,14,17)(10,13,15,23)(12,21,20,18)(16,24,19,22)(25,27)(26,33)(28,39)(29,38)(30,36)(31,32)(34,40)(35,37);(1,4,6,3)(2,5,7,8)(9,12,14,20)(10,16,15,19)(11,18,17,21)(13,22,23,24)(25,28,30,35)(26,34,32,29)(27,37,36,39)(31,40,33,38);(1,5)(2,3)(4,7)(6,8)(9,13)(10,17)(11,15)(12,22)(14,23)(16,21)(18,19)(20,24)(25,29,30,34)(26,35,32,28)(27,38,36,40)(31,39,33,37)Polycyclic presentation:
2^9|a^2=g,b^2=h,d^2=g,e^2=gi,f^2=h,b^a=bg,c^a=ch,d^a=di,e^a=egi,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,d^c=dgi,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, 8841934);