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 | #49741 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4109 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,3)(4,7,8,6)(9,10,16,19)(11,20,13,17)(12,15,14,18)(21,24,22,23)(26,30)(28,35)(31,36)(34,39);(2,3)(6,7)(9,11,16,13)(10,17,19,20)(12,21,14,22)(15,23,18,24)(27,33)(31,36)(34,39)(38,40);(1,3,5,2)(4,7,8,6)(9,12,16,14)(10,18,19,15)(11,21,13,22)(17,24,20,23)(25,26)(27,31)(28,32)(29,30)(33,36)(34,38)(35,37)(39,40);(1,4,5,8)(2,6,3,7)(9,13,16,11)(10,17,19,20)(12,21,14,22)(15,24,18,23);(9,14,16,12)(10,18,19,15)(11,22,13,21)(17,24,20,23)(25,27)(26,31)(28,34)(29,33)(30,36)(32,38)(35,39)(37,40);(9,15,16,18)(10,12,19,14)(11,23,13,24)(17,21,20,22)(25,28,29,35)(26,32,30,37)(27,34,33,39)(31,38,36,40)Polycyclic presentation:
2^9|a^2=g,b^2=gi,c^2=g,d^2=g,e^2=gi,f^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, 8839341);