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×#525 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4350 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,5)(3,7)(9,10)(11,18)(12,20)(13,17)(14,15)(16,21)(19,22)(23,24)(25,26)(27,30)(28,36)(29,32)(31,37)(33,35)(34,39)(38,40);(1,2,4,5)(3,6,7,8)(10,15)(11,17)(12,21)(22,24)(25,26)(27,33)(28,36)(29,32)(30,35)(31,37)(34,38)(39,40);(1,2)(3,6)(4,5)(7,8)(10,15)(13,18)(16,20)(22,24)(25,26,29,32)(27,30,35,33)(28,31,37,36)(34,38,40,39);(1,2)(3,6)(4,5)(7,8)(9,11,14,17)(10,13,15,18)(12,19,21,23)(16,22,20,24)(25,27)(26,30)(28,34)(29,35)(31,38)(32,33)(36,39)(37,40);(1,3,4,7)(2,6,5,8)(9,12)(10,16)(11,19)(13,22)(14,21)(15,20)(17,23)(18,24)(25,28)(26,31)(27,34)(29,37)(30,38)(32,36)(33,39)(35,40);(9,13)(10,17)(11,15)(12,22)(14,18)(16,23)(19,20)(21,24)(25,26)(27,30)(28,31)(29,32)(33,35)(34,38)(36,37)(39,40)Polycyclic presentation:
2^9|b^2=ghi,c^2=gh,d^2=hi,e^2=ghi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,f^c=fgi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8135751);