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 | #34829 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4510 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,3)(4,7,8,6)(12,20)(16,23)(19,24)(21,22)(26,31)(27,36)(34,37)(35,40);(2,3)(6,7)(9,10,14,18)(11,13,17,15)(12,16,20,23)(19,21,24,22)(25,26,30,31)(27,32,36,33)(28,34,38,37)(29,35,39,40);(1,3,5,2)(4,7,8,6)(9,11)(10,15)(12,19)(13,18)(14,17)(16,22)(20,24)(21,23)(25,27)(26,32)(28,35)(29,37)(30,36)(31,33)(34,39)(38,40);(1,4)(2,6)(3,7)(5,8)(9,11)(10,13)(12,19)(14,17)(15,18)(16,21)(20,24)(22,23)(25,27)(26,33)(28,35)(29,34)(30,36)(31,32)(37,39)(38,40);(9,12)(10,16)(11,19)(13,21)(14,20)(15,22)(17,24)(18,23)(25,28,30,38)(26,34,31,37)(27,35,36,40)(29,32,39,33);(9,13)(10,17)(11,18)(12,21)(14,15)(16,24)(19,23)(20,22)(25,29)(26,35)(27,37)(28,32)(30,39)(31,40)(33,38)(34,36)Polycyclic presentation:
2^9|a^2=gh,b^2=gi,c^2=gh,e^2=h,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, 8342010);