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 | #52006 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4378 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)(4,8)(6,7)(9,10,15,18)(11,13,21,16)(12,14,17,24)(19,20,22,23)(25,26,30,35)(27,32,36,31)(28,37,39,33)(29,38,40,34);(1,3)(2,5)(4,6)(7,8)(9,11,15,21)(10,16,18,13)(12,22,17,19)(14,20,24,23)(25,26)(27,31)(28,33)(29,34)(30,35)(32,36)(37,39)(38,40);(1,2)(3,8)(4,5)(6,7)(12,17)(14,24)(19,22)(20,23)(25,27,30,36)(26,31,35,32)(28,34,39,38)(29,37,40,33);(2,7)(3,4)(9,12,15,17)(10,14,18,24)(11,19,21,22)(13,20,16,23)(25,27)(26,32)(28,38)(29,33)(30,36)(31,35)(34,39)(37,40);(1,4,6,3)(2,5,7,8)(9,13,15,16)(10,11,18,21)(12,23,17,20)(14,22,24,19)(25,28,30,39)(26,33,35,37)(27,34,36,38)(29,32,40,31);(1,5)(2,3)(4,7)(6,8)(9,14)(10,17)(11,20)(12,18)(13,22)(15,24)(16,19)(21,23)(25,29,30,40)(26,34,35,38)(27,33,36,37)(28,32,39,31)Polycyclic presentation:
2^9|a^2=g,b^2=gh,c^2=h,d^2=gh,e^2=gi,f^2=h,b^a=bg,c^a=ch,d^a=di,e^a=egi,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,d^c=dgi,f^c=fgi,e^d=egi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8842005);