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 | #44595 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4273 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)(3,4)(5,7)(6,8)(9,10,14,15)(11,17,18,12)(13,21,22,16)(19,20,24,23)(25,26,32,36)(27,33,30,35)(28,34,29,31)(37,38,39,40);(1,3)(2,6)(4,7)(5,8)(9,11,14,18)(10,12,15,17)(13,19,22,24)(16,20,21,23)(25,27,32,30)(26,33,36,35)(28,39,29,37)(31,38,34,40);(10,15)(11,18)(16,21)(19,24)(25,28,32,29)(26,34,36,31)(27,37,30,39)(33,38,35,40);(2,7)(4,6)(9,10)(11,12)(13,16)(14,15)(17,18)(19,20)(21,22)(23,24)(25,29,32,28)(26,31,36,34)(27,37,30,39)(33,38,35,40);(1,4,5,6)(2,3,7,8)(9,12)(10,11)(13,20)(14,17)(15,18)(16,19)(21,24)(22,23)(25,30,32,27)(26,35,36,33)(28,39,29,37)(31,38,34,40);(1,3)(2,4)(5,8)(6,7)(9,13)(10,16)(11,19)(12,20)(14,22)(15,21)(17,23)(18,24)(25,31)(26,28)(27,38)(29,36)(30,40)(32,34)(33,39)(35,37)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=hi,d^2=hi,e^2=gi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^d=egi,f^e=fgi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10090313);