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 | |
4450 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)(11,18)(12,13)(14,15)(16,21)(17,22)(19,20)(23,24)(25,26,31,34)(27,30,33,40)(28,29,32,39)(35,36,37,38);(2,3)(6,7)(10,15)(12,22)(16,18)(19,23)(25,27)(26,30)(28,37)(29,38)(31,33)(32,35)(34,40)(36,39);(1,3,5,2)(4,7,8,6)(10,15)(13,17)(16,18)(20,24)(27,33)(30,40)(35,37)(36,38);(1,4,5,8)(2,6,3,7)(9,11,14,21)(10,16,15,18)(12,23,22,19)(13,20,17,24)(25,28)(26,29)(27,35)(30,36)(31,32)(33,37)(34,39)(38,40);(9,12)(10,17)(11,19)(13,15)(14,22)(16,20)(18,24)(21,23)(25,29)(26,32)(27,36)(28,34)(30,37)(31,39)(33,38)(35,40);(9,13,14,17)(10,12,15,22)(11,20,21,24)(16,23,18,19)(25,30,31,40)(26,33,34,27)(28,36,32,38)(29,37,39,35)Polycyclic presentation:
2^9|a^2=i,c^2=gh,d^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,e^b=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, 8306680);