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 | #44770 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4311 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,3)(4,7,8,6)(9,10)(11,12)(13,17)(14,15)(16,18)(19,20)(21,22)(23,24);(2,3)(6,7)(9,10)(11,18)(12,16)(13,17)(14,15)(19,22)(20,21)(23,24)(25,26,31,35)(27,36,37,32)(28,29,33,39)(30,34,38,40);(1,3,5,2)(4,7,8,6)(10,15)(11,16)(17,24)(19,21)(25,27,31,37)(26,32,35,36)(28,38,33,30)(29,34,39,40);(1,4,5,8)(2,6,3,7)(9,11,14,16)(10,12,15,18)(13,21,23,19)(17,22,24,20)(25,28,31,33)(26,29,35,39)(27,30,37,38)(32,40,36,34);(9,12)(10,16)(11,15)(13,22)(14,18)(17,19)(20,23)(21,24)(25,29)(26,33)(27,34)(28,35)(30,32)(31,39)(36,38)(37,40);(9,13,14,23)(10,17,15,24)(11,19,16,21)(12,20,18,22)(25,30)(26,34)(27,28)(29,36)(31,38)(32,39)(33,37)(35,40)Polycyclic presentation:
2^9|a^2=g,b^2=ghi,c^2=hi,d^2=i,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8627404);