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 | C2×#121 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4150 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,3)(4,7,8,6)(9,10)(11,19)(12,14)(13,16)(15,17)(18,23)(20,21)(22,24)(25,26,31,34)(27,30,33,39)(28,29,32,38)(35,36,37,40);(2,3)(6,7)(9,11,15,13)(10,16,17,19)(12,22,23,20)(14,21,18,24)(25,27)(26,30)(28,35)(29,36)(31,33)(32,37)(34,39)(38,40);(1,3,5,2)(4,7,8,6)(10,17)(14,18)(16,19)(21,24)(27,33)(30,39)(35,37)(36,40);(1,4,5,8)(2,6,3,7)(9,12)(10,18)(11,20)(13,22)(14,17)(15,23)(16,21)(19,24)(25,28)(26,29)(27,35)(30,36)(31,32)(33,37)(34,38)(39,40);(9,13,15,11)(10,19,17,16)(12,20,23,22)(14,24,18,21)(25,29)(26,32)(27,36)(28,34)(30,37)(31,38)(33,40)(35,39);(9,14,15,18)(10,12,17,23)(11,21,13,24)(16,22,19,20)(25,30,31,39)(26,33,34,27)(28,36,32,40)(29,37,38,35)Polycyclic presentation:
2^9|a^2=i,b^2=h,c^2=gh,d^2=gh,e^2=h,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,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, 8231299);