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 | #42578 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4437 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)(11,12,16,15)(13,17,22,21)(19,20,24,23)(25,26)(27,32)(28,29)(30,34)(31,35)(33,38)(36,39)(37,40);(1,2)(3,8)(4,6)(5,7)(9,11,14,16)(10,15,18,12)(13,19,22,24)(17,23,21,20)(27,35)(31,32)(36,40)(37,39);(1,3,5,4)(2,6,7,8)(25,27)(26,31)(28,36)(29,37)(30,35)(32,34)(33,39)(38,40);(1,2)(3,6)(4,8)(5,7)(9,10)(11,15)(12,16)(13,21)(14,18)(17,22)(19,20)(23,24)(25,27)(26,32)(28,36)(29,39)(30,35)(31,34)(33,37)(38,40);(9,12,14,15)(10,16,18,11)(13,20,22,23)(17,24,21,19)(25,28)(26,33)(27,36)(29,34)(30,38)(31,39)(32,37)(35,40);(1,4,5,3)(2,6,7,8)(9,13,14,22)(10,17,18,21)(11,19,16,24)(12,20,15,23)(25,29,30,33)(26,28,34,38)(27,37,35,39)(31,36,32,40)Polycyclic presentation:
2^9|a^2=g,b^2=g,c^2=hi,e^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^d=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8336462);