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 | |
4404 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)(3,5)(4,7)(6,8)(10,12)(15,18)(16,20)(22,24)(25,26)(27,33)(28,29)(30,34)(31,32)(35,38)(36,37)(39,40);(2,3)(6,7)(11,17)(15,18)(19,23)(22,24)(26,32)(28,33)(34,40)(36,38);(1,2,5,3)(4,6,8,7)(9,10,14,12)(11,15,17,18)(13,16,21,20)(19,22,23,24)(25,27,31,29)(26,33,32,28)(30,35,39,37)(34,38,40,36);(1,3,5,2)(4,7,8,6)(9,11,14,17)(10,15,12,18)(13,19,21,23)(16,22,20,24)(25,28)(26,27)(29,32)(30,38)(31,33)(34,37)(35,40)(36,39);(1,4)(2,6)(3,7)(5,8)(9,12,14,10)(11,18,17,15)(13,20,21,16)(19,24,23,22)(25,29,31,27)(26,28,32,33)(30,35,39,37)(34,38,40,36);(9,13,14,21)(10,16,12,20)(11,19,17,23)(15,22,18,24)(25,30,31,39)(26,34,32,40)(27,35,29,37)(28,36,33,38)Polycyclic presentation:
2^9|c^2=ghi,d^2=h,e^2=gh,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^d=fgi,f^e=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8602404);