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 | |
4504 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10)(11,18)(12,16)(13,17)(14,15)(19,24)(20,23)(21,22)(26,31)(27,28)(33,37)(34,36);(1,2,5,3)(4,6,8,7)(10,15)(11,17)(16,23)(19,21)(25,26)(27,32)(28,35)(29,37)(30,31)(33,38)(34,40)(36,39);(2,3)(6,7)(12,20)(16,23)(19,21)(22,24)(25,27,30,28)(26,32,31,35)(29,36,38,34)(33,40,37,39);(2,3)(4,8)(9,11)(10,13)(12,21)(14,17)(15,18)(16,24)(19,20)(22,23)(25,28,30,27)(26,32,31,35)(29,34,38,36)(33,40,37,39);(1,3,5,2)(4,7,8,6)(9,12)(10,16)(11,19)(13,22)(14,20)(15,23)(17,21)(18,24)(25,29,30,38)(26,33,31,37)(27,34,28,36)(32,39,35,40);(1,4)(2,6)(3,7)(5,8)(9,13)(10,17)(11,15)(12,22)(14,18)(16,21)(19,23)(20,24)(25,26)(27,32)(28,35)(29,33)(30,31)(34,39)(36,40)(37,38)Polycyclic presentation:
2^9|b^2=i,c^2=h,d^2=h,e^2=hi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,e^c=egi,e^d=egi,f^d=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9012485);