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 | |
4241 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)(3,6)(4,5)(7,8)(9,10)(11,18)(12,16)(13,21)(14,15)(17,22)(19,20)(23,24);(1,3,5,8)(2,6,4,7)(10,15)(12,18)(17,21)(20,24)(25,26,30,31)(27,32,36,39)(28,29,33,38)(34,40,37,35);(1,3)(2,7)(4,6)(5,8)(10,15)(11,16)(13,22)(20,24)(26,31)(29,38)(32,39)(35,40);(1,3,5,8)(2,6,4,7)(9,11)(10,12)(13,19)(14,16)(15,18)(17,20)(21,24)(22,23)(25,27)(26,32)(28,37)(29,35)(30,36)(31,39)(33,34)(38,40);(1,3)(2,6)(4,7)(5,8)(9,12)(10,16)(11,15)(13,20)(14,18)(17,23)(19,21)(22,24)(25,28,30,33)(26,29,31,38)(27,34,36,37)(32,40,39,35);(1,4,5,2)(3,7,8,6)(9,13)(10,17)(11,19)(12,20)(14,22)(15,21)(16,23)(18,24)(25,29)(26,33)(27,35)(28,31)(30,38)(32,34)(36,40)(37,39)Polycyclic presentation:
2^9|b^2=i,d^2=gh,e^2=ghi,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,f^c=fgi,e^d=egi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9360253);