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 | |
4467 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,3)(4,8)(11,17)(12,20)(15,18)(16,21)(25,26,31,34)(27,29,37,32)(28,33,39,30)(35,40,38,36);(1,2,5,3)(4,6,8,7)(9,10)(11,18)(12,21)(13,14)(15,17)(16,20)(19,23)(22,24)(25,27,31,37)(26,32,34,29)(28,38,39,35)(30,36,33,40);(1,2,5,3)(4,7,8,6)(10,14)(11,17)(16,21)(19,22)(28,39)(30,33)(35,38)(36,40);(2,3)(6,7)(9,11)(10,15)(12,22)(13,17)(14,18)(16,24)(19,20)(21,23)(25,28)(26,33)(27,35)(29,40)(30,34)(31,39)(32,36)(37,38);(1,3,5,2)(4,7,8,6)(9,12)(10,16)(11,19)(13,20)(14,21)(15,23)(17,22)(18,24)(25,29,31,32)(26,27,34,37)(28,36,39,40)(30,35,33,38);(1,4,5,8)(2,6,3,7)(9,10)(11,15)(12,16)(13,14)(17,18)(19,23)(20,21)(22,24)(25,30,31,33)(26,28,34,39)(27,36,37,40)(29,35,32,38)Polycyclic presentation:
2^9|a^2=gh,b^2=g,c^2=h,e^2=g,f^2=g,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, 8840284);