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 | |
4590 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2)(3,8)(4,6)(5,7)(9,10,14,15)(11,18,17,13)(12,16,20,24)(19,22,21,23)(25,26,31,28)(27,29,32,35)(30,36,39,33)(34,38,40,37);(1,3,5,4)(2,6,7,8)(10,15)(11,17)(12,20)(22,23)(25,27)(26,29)(28,35)(30,34)(31,32)(33,37)(36,38)(39,40);(10,15)(11,17)(16,24)(19,21)(26,28)(27,32)(33,36)(34,40);(1,2,5,7)(3,8,4,6)(9,11,14,17)(10,13,15,18)(12,21,20,19)(16,22,24,23)(25,28,31,26)(27,29,32,35)(30,36,39,33)(34,37,40,38);(1,2)(3,6)(4,8)(5,7)(9,12)(10,16)(11,19)(13,23)(14,20)(15,24)(17,21)(18,22)(25,29)(26,32)(27,28)(30,38)(31,35)(33,34)(36,40)(37,39);(1,4,5,3)(2,6,7,8)(9,13)(10,17)(11,15)(12,22)(14,18)(16,19)(20,23)(21,24)(25,30,31,39)(26,33,28,36)(27,34,32,40)(29,37,35,38)Polycyclic presentation:
2^9|a^2=h,b^2=ghi,d^2=gi,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,d^c=dh,e^d=egi,f^e=fgi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10154683);