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 | |
4410 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2)(3,5)(4,7)(6,8)(9,10,14,16)(11,17,18,13)(12,15,20,23)(19,24,21,22)(25,26,31,34)(27,32,30,28)(29,33,39,40)(35,37,38,36);(1,2,6,8)(3,5,7,4)(9,10)(11,13)(12,15)(14,16)(17,18)(19,22)(20,23)(21,24)(25,27,31,30)(26,32,34,28)(29,35,39,38)(33,37,40,36);(1,3,6,7)(2,4,8,5)(11,18)(12,20)(13,17)(15,23)(25,28,31,32)(26,27,34,30)(29,37,39,36)(33,38,40,35);(1,4,6,5)(2,3,8,7)(9,11)(10,13)(12,21)(14,18)(15,24)(16,17)(19,20)(22,23)(25,28)(26,27)(29,37)(30,34)(31,32)(33,38)(35,40)(36,39);(1,2)(3,4)(5,7)(6,8)(9,12)(10,15)(11,19)(13,22)(14,20)(16,23)(17,24)(18,21)(25,29)(26,33)(27,35)(28,36)(30,38)(31,39)(32,37)(34,40);(1,5,6,4)(2,7,8,3)(9,13)(10,11)(12,22)(14,17)(15,19)(16,18)(20,24)(21,23)(25,30,31,27)(26,28,34,32)(29,38,39,35)(33,36,40,37)Polycyclic presentation:
2^9|a^2=gi,b^2=i,c^2=i,d^2=h,f^2=i,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, 8936297);