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 | |
4527 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)(3,6)(4,7)(5,8)(9,10,15,18)(11,17,13,16)(12,21,23,14)(19,20,22,24)(28,36)(33,37)(34,40)(38,39);(3,8)(5,6)(9,11,15,13)(10,16,18,17)(12,19,23,22)(14,20,21,24)(25,26,30,29)(27,32,35,31)(28,33,36,37)(34,39,40,38);(1,3)(2,5)(4,8)(6,7)(25,27)(26,31)(28,34)(29,32)(30,35)(33,38)(36,40)(37,39);(1,3)(2,6)(4,8)(5,7)(9,12,15,23)(10,14,18,21)(11,19,13,22)(16,20,17,24)(25,27)(26,32)(28,34)(29,31)(30,35)(33,39)(36,40)(37,38);(9,13,15,11)(10,17,18,16)(12,22,23,19)(14,24,21,20)(25,28,30,36)(26,33,29,37)(27,34,35,40)(31,38,32,39);(1,2)(3,5)(4,7)(6,8)(9,14)(10,12)(11,20)(13,24)(15,21)(16,19)(17,22)(18,23)(25,29,30,26)(27,32,35,31)(28,37,36,33)(34,39,40,38)Polycyclic presentation:
2^9|a^2=g,b^2=ghi,d^2=g,e^2=ghi,f^2=hi,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,f^d=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8369260);