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 | |
4609 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)(16,19)(17,22)(20,21)(23,24)(25,26,31,30)(27,32,34,29)(28,33,37,38)(35,39,40,36);(2,3)(6,7)(10,15)(11,19)(12,22)(21,23)(25,27)(26,32)(28,35)(29,30)(31,34)(33,39)(36,38)(37,40);(1,3,5,2)(4,7,8,6)(10,15)(12,22)(16,18)(20,24)(27,34)(29,32)(35,40)(36,39);(1,4,5,8)(2,6,3,7)(9,11,14,19)(10,16,15,18)(12,20,22,24)(13,23,17,21)(25,28)(26,33)(27,35)(29,36)(30,38)(31,37)(32,39)(34,40);(9,12)(10,17)(11,20)(13,15)(14,22)(16,21)(18,23)(19,24)(25,29)(26,27)(28,36)(30,34)(31,32)(33,35)(37,39)(38,40);(9,13,14,17)(10,12,15,22)(11,21,19,23)(16,24,18,20)(25,30,31,26)(27,29,34,32)(28,33,37,38)(35,39,40,36)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=dh,e^b=egi,e^c=egi,f^d=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8541323);