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 | |
4838 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,3)(4,7,8,6)(9,10,15,17)(11,14,12,16)(13,18,22,24)(19,21,20,23)(25,26,30,32)(27,34,37,31)(28,33,38,36)(29,39,40,35);(2,3)(6,7)(9,11,15,12)(10,16,17,14)(13,20,22,19)(18,21,24,23)(25,27)(26,31)(28,35)(29,36)(30,37)(32,34)(33,40)(38,39);(1,3,5,2)(4,7,8,6)(10,17)(11,12)(18,24)(19,20)(26,32)(28,38)(31,34)(35,39);(1,4,5,8)(2,6,3,7)(9,12,15,11)(10,14,17,16)(13,20,22,19)(18,21,24,23)(25,28,30,38)(26,33,32,36)(27,35,37,39)(29,31,40,34);(9,13)(10,18)(11,19)(12,20)(14,21)(15,22)(16,23)(17,24)(25,27)(26,34)(28,35)(29,33)(30,37)(31,32)(36,40)(38,39);(9,14)(10,11)(12,17)(13,21)(15,16)(18,19)(20,24)(22,23)(25,29)(26,35)(27,36)(28,31)(30,40)(32,39)(33,37)(34,38)Polycyclic presentation:
2^9|a^2=g,b^2=ghi,c^2=gh,d^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dh,f^e=fgi77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9820417);