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 | |
4491 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,8)(3,6,4,7)(9,10,14,13)(11,16,18,15)(12,20,22,17)(19,23,21,24)(25,26)(27,33)(28,31)(29,34)(30,32)(35,40)(36,38)(37,39);(1,2)(3,6)(4,7)(5,8)(9,11,14,18)(10,15,13,16)(12,21,22,19)(17,24,20,23)(25,27,30,28)(26,31,32,33)(29,35,37,36)(34,38,39,40);(10,13)(11,18)(17,20)(19,21)(26,32)(31,33)(34,39)(38,40);(9,11)(10,16)(12,19)(13,15)(14,18)(17,24)(20,23)(21,22)(25,26)(27,31)(28,33)(29,34)(30,32)(35,38)(36,40)(37,39);(1,3,5,4)(2,6,8,7)(9,12,14,22)(10,17,13,20)(11,19,18,21)(15,23,16,24)(25,28,30,27)(26,33,32,31)(29,36,37,35)(34,40,39,38);(1,4,5,3)(2,7,8,6)(9,13,14,10)(11,15,18,16)(12,20,22,17)(19,23,21,24)(25,29,30,37)(26,34,32,39)(27,35,28,36)(31,38,33,40)Polycyclic presentation:
2^9|a^2=ghi,b^2=h,e^2=g,f^2=g,b^a=bg,c^a=ch,e^a=eh,c^b=ci,d^b=di,e^b=ei,f^b=fi,d^c=dh,f^c=fi104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7712264);