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 | |
4441 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)(3,7)(4,8)(5,6)(9,10)(11,18)(12,13)(14,15)(16,19)(17,22)(20,21)(23,24)(26,28)(31,34)(32,36)(38,40);(2,6)(3,8)(10,15)(11,19)(12,22)(21,23)(27,33)(31,34)(35,39)(38,40);(1,3)(2,4)(5,8)(6,7)(10,15)(12,22)(16,18)(20,24)(25,26,30,28)(27,31,33,34)(29,32,37,36)(35,38,39,40);(1,3)(2,4)(5,8)(6,7)(9,11)(10,16)(12,20)(13,23)(14,19)(15,18)(17,21)(22,24)(25,27)(26,31)(28,34)(29,35)(30,33)(32,38)(36,40)(37,39);(1,4)(2,3)(5,7)(6,8)(9,12)(10,17)(11,20)(13,15)(14,22)(16,21)(18,23)(19,24)(25,28,30,26)(27,34,33,31)(29,36,37,32)(35,40,39,38);(9,13,14,17)(10,12,15,22)(11,21,19,23)(16,24,18,20)(25,29,30,37)(26,32,28,36)(27,35,33,39)(31,38,34,40)Polycyclic presentation:
2^9|c^2=gh,e^2=gh,f^2=hi,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, 8541080);