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 | #37804 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4185 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)(11,18)(12,15)(13,16)(14,17)(19,21)(20,23)(22,24)(25,26,30,31)(27,29,34,32)(28,36,38,33)(35,39,40,37);(1,2)(3,7)(4,5)(6,8)(9,11)(10,12)(13,19)(14,15)(16,20)(17,18)(21,24)(22,23)(26,31)(27,34)(33,36)(35,40);(2,5)(6,7)(9,12,14,18)(10,15,17,11)(13,21,22,20)(16,19,24,23)(25,27,30,34)(26,32,31,29)(28,35,38,40)(33,39,36,37);(3,8)(6,7)(9,11)(10,12)(13,19)(14,15)(16,20)(17,18)(21,24)(22,23)(25,26)(27,32)(28,33)(29,34)(30,31)(35,39)(36,38)(37,40);(1,3,4,8)(2,6,5,7)(9,10)(11,18)(12,15)(13,16)(14,17)(19,21)(20,23)(22,24)(25,28)(26,33)(27,35)(29,37)(30,38)(31,36)(32,39)(34,40);(1,2)(3,6)(4,5)(7,8)(9,13,14,22)(10,16,17,24)(11,19,15,23)(12,20,18,21)(25,29,30,32)(26,34,31,27)(28,37,38,39)(33,40,36,35)Polycyclic presentation:
2^9|a^2=h,c^2=g,e^2=hi,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,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, 8268852);