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 | |
4303 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)(11,14)(12,15)(13,18)(16,22)(17,19)(20,24)(21,23)(25,26,30,33)(27,31,35,29)(28,32,36,39)(34,37,40,38);(2,3)(6,7)(9,10)(11,14)(12,17)(13,16)(15,19)(18,22)(20,24)(21,23)(25,27,30,35)(26,29,33,31)(28,34,36,40)(32,38,39,37);(1,3,5,2)(4,7,8,6)(10,11)(12,19)(16,18)(20,21)(28,36)(32,39)(34,40)(37,38);(1,4,5,8)(2,6,3,7)(9,11,14,10)(12,17,19,15)(13,18,22,16)(20,24,21,23)(25,27)(26,31)(28,34)(29,33)(30,35)(32,37)(36,40)(38,39);(9,12,14,19)(10,15,11,17)(13,21,22,20)(16,24,18,23)(25,28,30,36)(26,32,33,39)(27,34,35,40)(29,38,31,37);(9,13)(10,16)(11,18)(12,20)(14,22)(15,23)(17,24)(19,21)(25,29,30,31)(26,27,33,35)(28,37,36,38)(32,40,39,34)Polycyclic presentation:
2^9|a^2=g,b^2=ghi,c^2=hi,d^2=i,e^2=gi,f^2=ghi,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,d^c=dh,e^c=egi,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9993556);