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 | |
4546 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,4)(3,7,8,6)(9,10)(11,18)(12,15)(13,16)(14,17)(19,24)(20,22)(21,23)(25,26)(27,31)(28,33)(29,37)(30,32)(34,38)(35,36)(39,40);(1,3)(2,6)(4,7)(5,8)(9,11,15,13)(10,16,12,18)(14,19,22,21)(17,23,20,24)(25,27,30,33)(26,31,32,28)(29,35,38,40)(34,39,37,36);(2,4)(6,7)(14,22)(17,20)(19,21)(23,24)(26,32)(27,33)(34,37)(35,40);(1,3)(2,6)(4,7)(5,8)(9,12,15,10)(11,16,13,18)(14,20,22,17)(19,23,21,24)(25,28,30,31)(26,33,32,27)(29,36,38,39)(34,40,37,35);(1,3)(2,6)(4,7)(5,8)(9,13,15,11)(10,16,12,18)(14,21,22,19)(17,23,20,24)(25,29,30,38)(26,34,32,37)(27,35,33,40)(28,36,31,39);(1,4,5,2)(3,7,8,6)(9,14,15,22)(10,17,12,20)(11,19,13,21)(16,23,18,24)(25,26)(27,31)(28,33)(29,34)(30,32)(35,39)(36,40)(37,38)Polycyclic presentation:
2^9|a^2=hi,b^2=gh,d^2=gh,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^c=fgi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8086878);