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 | |
4556 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)(3,7)(4,5)(6,8)(9,10)(11,17)(12,20)(13,14)(15,18)(16,22)(19,23)(21,24)(25,26,31,36)(27,34,29,32)(28,33,39,37)(30,38,40,35);(2,5)(6,7)(10,14)(15,17)(16,20)(23,24)(27,29)(30,40)(32,34)(35,38);(1,2)(3,6)(4,5)(7,8)(11,18)(15,17)(19,21)(23,24)(25,27,31,29)(26,32,36,34)(28,38,39,35)(30,37,40,33);(1,3,4,8)(2,6,5,7)(9,11)(10,15)(12,21)(13,18)(14,17)(16,24)(19,22)(20,23)(25,28)(26,33)(27,35)(29,38)(30,34)(31,39)(32,40)(36,37);(9,12)(10,16)(11,19)(13,22)(14,20)(15,23)(17,24)(18,21)(25,29,31,27)(26,34,36,32)(28,35,39,38)(30,33,40,37);(9,10)(11,15)(12,16)(13,14)(17,18)(19,23)(20,22)(21,24)(25,30,31,40)(26,35,36,38)(27,37,29,33)(28,34,39,32)Polycyclic presentation:
2^9|a^2=h,c^2=h,d^2=hi,e^2=h,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^d=egi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8746325);