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 | |
4572 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,6)(4,8)(9,10,15,13)(11,12,19,16)(14,21,22,17)(18,23,24,20);(2,6)(3,7)(9,11,15,19)(10,16,13,12)(14,18,22,24)(17,23,21,20)(25,26,30,31)(27,32,35,40)(28,33,38,37)(29,34,39,36);(1,2,5,6)(3,8,7,4)(10,13)(12,16)(14,22)(18,24)(26,31)(27,35)(28,38)(29,39);(2,6)(3,7)(9,12)(10,11)(13,19)(14,20)(15,16)(17,18)(21,24)(22,23)(25,27)(26,32)(28,34)(29,37)(30,35)(31,40)(33,39)(36,38);(1,3,5,7)(2,4,6,8)(9,13,15,10)(11,16,19,12)(14,21,22,17)(18,23,24,20)(25,28)(26,33)(27,36)(29,32)(30,38)(31,37)(34,35)(39,40);(1,4)(2,7)(3,6)(5,8)(9,14)(10,17)(11,18)(12,20)(13,21)(15,22)(16,23)(19,24)(25,29)(26,34)(27,37)(28,32)(30,39)(31,36)(33,35)(38,40)Polycyclic presentation:
2^9|a^2=g,b^2=hi,c^2=gh,e^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8884565);