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 | #50733 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4283 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)(3,5)(4,7)(6,8)(9,10,14,16)(11,15,19,23)(12,17,18,13)(20,21,22,24)(25,26,30,29)(27,31,33,35)(28,32,36,37)(34,38,39,40);(1,2,6,8)(3,5,7,4)(9,11)(10,15)(12,22)(13,21)(14,19)(16,23)(17,24)(18,20)(25,27)(26,31)(28,34)(29,35)(30,33)(32,38)(36,39)(37,40);(1,3,6,7)(2,4,8,5)(10,16)(11,19)(13,17)(20,22)(27,33)(28,36)(31,35)(32,37);(1,4,6,5)(2,3,8,7)(9,11)(10,15)(12,22)(13,21)(14,19)(16,23)(17,24)(18,20)(25,28,30,36)(26,32,29,37)(27,34,33,39)(31,38,35,40);(1,2)(3,4)(5,7)(6,8)(9,12,14,18)(10,17,16,13)(11,20,19,22)(15,21,23,24)(25,29,30,26)(27,35,33,31)(28,32,36,37)(34,38,39,40);(1,5,6,4)(2,7,8,3)(9,13,14,17)(10,18,16,12)(11,21,19,24)(15,20,23,22)(25,27)(26,31)(28,34)(29,35)(30,33)(32,38)(36,39)(37,40)Polycyclic presentation:
2^9|a^2=gi,b^2=g,c^2=g,d^2=ghi,e^2=gi,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,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, 8829950);