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 | #45623 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4358 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,5)(3,7)(9,10,14,18)(11,16,12,15)(13,17,22,21)(19,24,20,23)(27,35)(28,37)(31,38)(32,40);(1,2,4,5)(3,6,7,8)(9,11,14,12)(10,15,18,16)(13,19,22,20)(17,23,21,24)(25,26,30,34)(27,31,35,38)(28,32,37,40)(29,33,36,39);(1,2)(3,6)(4,5)(7,8)(25,27)(26,31)(28,36)(29,37)(30,35)(32,39)(33,40)(34,38);(1,2)(3,6)(4,5)(7,8)(9,10)(11,15)(12,16)(13,21)(14,18)(17,22)(19,24)(20,23)(25,26)(27,31)(28,32)(29,33)(30,34)(35,38)(36,39)(37,40);(1,3,4,7)(2,6,5,8)(9,12,14,11)(10,16,18,15)(13,20,22,19)(17,24,21,23)(25,28,30,37)(26,32,34,40)(27,36,35,29)(31,39,38,33);(9,13)(10,17)(11,19)(12,20)(14,22)(15,23)(16,24)(18,21)(25,29,30,36)(26,33,34,39)(27,37,35,28)(31,40,38,32)Polycyclic presentation:
2^9|a^2=gi,b^2=ghi,e^2=ghi,f^2=hi,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,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, 8370852);