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 | |
4136 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10,14,19)(11,15,21,16)(12,17,22,23)(13,20,24,18)(25,26)(27,33)(28,36)(29,34)(30,31)(32,38)(35,37)(39,40);(1,2)(3,6)(4,7)(5,8)(9,11)(10,15)(12,18)(13,17)(14,21)(16,19)(20,22)(23,24)(26,31)(27,34)(32,36)(35,40);(1,3)(2,5)(4,8)(6,7)(9,11,14,21)(10,16,19,15)(12,20,22,18)(13,23,24,17)(25,27)(26,29)(28,35)(30,34)(31,33)(32,37)(36,39)(38,40);(3,8)(5,6)(9,11)(10,15)(12,18)(13,17)(14,21)(16,19)(20,22)(23,24)(25,28,30,38)(26,32,31,36)(27,35,34,40)(29,37,33,39);(1,3)(2,6)(4,8)(5,7)(9,12)(10,17)(11,20)(13,16)(14,22)(15,24)(18,21)(19,23)(25,26)(27,29)(28,32)(30,31)(33,34)(35,37)(36,38)(39,40);(1,2)(3,6)(4,7)(5,8)(9,13,14,24)(10,18,19,20)(11,17,21,23)(12,16,22,15)(25,29)(26,27)(28,37)(30,33)(31,34)(32,35)(36,40)(38,39)Polycyclic presentation:
2^9|a^2=gh,c^2=gh,d^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,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, 8811650);