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:
(9,10)(11,18)(12,13)(14,15)(16,21)(17,22)(19,20)(23,24)(25,26,31,29)(27,34,32,28)(30,33,38,37)(35,40,39,36);(1,2,5,6)(3,7,8,4)(10,15)(12,22)(16,18)(19,23)(25,27,31,32)(26,28,29,34)(30,35,38,39)(33,36,37,40);(2,6)(4,7)(10,15)(13,17)(16,18)(20,24)(30,38)(33,37)(35,39)(36,40);(2,6)(3,8)(9,11,14,21)(10,16,15,18)(12,23,22,19)(13,20,17,24)(25,28,31,34)(26,32,29,27)(30,36,38,40)(33,39,37,35);(1,3)(2,4)(5,8)(6,7)(9,12)(10,17)(11,19)(13,15)(14,22)(16,20)(18,24)(21,23)(25,29,31,26)(27,28,32,34)(30,37,38,33)(35,36,39,40);(1,4)(2,3)(5,7)(6,8)(9,13,14,17)(10,12,15,22)(11,20,21,24)(16,23,18,19)(25,30,31,38)(26,33,29,37)(27,35,32,39)(28,36,34,40)Polycyclic presentation:
2^9|a^2=gh,b^2=ghi,d^2=g,e^2=gh,f^2=g,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, 8295520);