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 | |
4438 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)(3,8)(4,7)(5,6)(10,15)(11,18)(12,20)(21,23)(26,31)(27,35)(28,38)(29,39);(2,6)(7,8)(9,10,14,15)(11,13,18,16)(12,17,20,22)(19,21,24,23)(25,26,30,31)(27,32,35,40)(28,37,38,33)(29,34,39,36);(9,11,14,18)(10,16,15,13)(12,19,20,24)(17,23,22,21)(25,27)(26,32)(28,36)(29,37)(30,35)(31,40)(33,39)(34,38);(1,3,5,4)(2,7,6,8)(9,10)(11,16)(12,17)(13,18)(14,15)(19,23)(20,22)(21,24)(25,26)(27,32)(28,33)(29,34)(30,31)(35,40)(36,39)(37,38);(1,2)(3,7)(4,8)(5,6)(9,12,14,20)(10,17,15,22)(11,19,18,24)(13,21,16,23)(25,28,30,38)(26,33,31,37)(27,36,35,34)(29,32,39,40);(1,4,5,3)(2,8,6,7)(9,13,14,16)(10,18,15,11)(12,21,20,23)(17,24,22,19)(25,29,30,39)(26,34,31,36)(27,37,35,33)(28,32,38,40)Polycyclic presentation:
2^9|b^2=h,c^2=i,d^2=gh,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,e^b=egi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8015972);