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 |
| prime radical | #34547 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4415 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(4,8)(6,7)(9,10,15,12)(11,16,17,13)(14,20,22,18)(19,21,24,23)(25,26,31,32)(27,33,28,30)(29,37,39,34)(35,38,36,40);(1,2,5,3)(4,6,8,7)(9,11,15,17)(10,16,12,13)(14,19,22,24)(18,23,20,21)(26,32)(27,28)(34,37)(35,36);(2,3)(6,7)(10,12)(11,17)(18,20)(19,24)(25,27,31,28)(26,30,32,33)(29,35,39,36)(34,38,37,40);(1,3,5,2)(4,7,8,6)(9,12,15,10)(11,16,17,13)(14,20,22,18)(19,23,24,21)(25,28,31,27)(26,33,32,30)(29,36,39,35)(34,40,37,38);(9,13,15,16)(10,17,12,11)(14,21,22,23)(18,24,20,19)(25,29,31,39)(26,34,32,37)(27,35,28,36)(30,38,33,40);(1,4)(2,6)(3,7)(5,8)(9,14)(10,18)(11,19)(12,20)(13,21)(15,22)(16,23)(17,24)(25,30,31,33)(26,27,32,28)(29,38,39,40)(34,35,37,36)Polycyclic presentation:
2^9|a^2=h,b^2=gi,c^2=g,d^2=i,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8014636);