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 |
| prime radical | #42606 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4445 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,6)(3,8,4,7)(9,10,14,18)(11,15,16,12)(13,21,22,17)(19,20,24,23)(25,26,30,31)(27,29,34,32)(28,36,38,33)(35,39,40,37);(2,6)(7,8)(9,11)(10,15)(12,18)(13,19)(14,16)(17,23)(20,21)(22,24)(26,31)(27,34)(33,36)(35,40);(2,6)(7,8)(9,12,14,15)(10,16,18,11)(13,20,22,23)(17,24,21,19)(25,26)(27,32)(28,33)(29,34)(30,31)(35,39)(36,38)(37,40);(1,2)(3,7)(4,8)(5,6)(9,12)(10,11)(13,20)(14,15)(16,18)(17,19)(21,24)(22,23)(25,27,30,34)(26,32,31,29)(28,35,38,40)(33,39,36,37);(1,3,5,4)(2,7,6,8)(9,13,14,22)(10,17,18,21)(11,19,16,24)(12,20,15,23)(25,28,30,38)(26,33,31,36)(27,35,34,40)(29,37,32,39);(1,4,5,3)(2,8,6,7)(9,11)(10,12)(13,19)(14,16)(15,18)(17,20)(21,23)(22,24)(25,29,30,32)(26,34,31,27)(28,37,38,39)(33,40,36,35)Polycyclic presentation:
2^9|a^2=h,c^2=gh,d^2=ghi,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8014482);