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 | |
4685 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)(9,10,15,20)(11,21,22,16)(12,13,18,17)(14,19,23,24);(2,6)(3,7)(9,11,15,22)(10,16,20,21)(12,19,18,24)(13,14,17,23)(25,26,30,31)(27,32,35,40)(28,33,38,37)(29,34,39,36);(1,2,5,6)(3,8,7,4)(12,18)(13,17)(14,23)(19,24)(26,31)(27,35)(28,38)(29,39);(2,6)(3,7)(9,12)(10,17)(11,19)(13,20)(14,21)(15,18)(16,23)(22,24)(25,27)(26,32)(28,34)(29,37)(30,35)(31,40)(33,39)(36,38);(1,3,5,7)(2,4,6,8)(9,13,15,17)(10,18,20,12)(11,14,22,23)(16,24,21,19)(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)(10,19)(11,17)(12,21)(13,22)(15,23)(16,18)(20,24)(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=ghi,c^2=h,e^2=gh,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, 8884579);