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 | #37130 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4204 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10,14,18)(11,13,17,15)(12,16,20,23)(19,21,24,22)(25,26,32,36)(27,37,39,33)(28,30,34,29)(31,35,40,38);(1,2)(3,7)(4,5)(6,8)(12,20)(16,23)(19,24)(21,22)(25,27,32,39)(26,33,36,37)(28,31,34,40)(29,35,30,38);(2,5)(6,7)(9,11,14,17)(10,15,18,13)(12,19,20,24)(16,22,23,21)(25,28)(26,29)(27,31)(30,36)(32,34)(33,35)(37,38)(39,40);(9,12)(10,16)(11,19)(13,21)(14,20)(15,22)(17,24)(18,23)(25,29,32,30)(26,34,36,28)(27,38,39,35)(31,37,40,33);(1,3,4,8)(2,6,5,7)(9,10)(11,15)(12,16)(13,17)(14,18)(19,22)(20,23)(21,24)(25,30,32,29)(26,28,36,34)(27,35,39,38)(31,33,40,37);(1,2)(3,6)(4,5)(7,8)(9,13,14,15)(10,17,18,11)(12,21,20,22)(16,24,23,19)(25,31)(26,35)(27,34)(28,39)(29,37)(30,33)(32,40)(36,38)Polycyclic presentation:
2^9|a^2=h,b^2=g,c^2=gh,d^2=g,e^2=gi,f^2=gh,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, 8122335);