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 | #35470 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4332 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)(11,16)(12,18)(13,21)(14,15)(17,22)(19,20)(23,24)(25,26,30,31)(27,29,33,39)(28,35,36,32)(34,40,37,38);(1,3)(2,5)(4,6)(7,8)(10,15)(11,18)(17,21)(19,24)(26,31)(27,33)(28,36)(38,40);(1,2)(3,8)(4,5)(6,7)(10,15)(11,18)(13,22)(20,23)(27,33)(29,39)(34,37)(38,40);(2,7)(3,4)(9,11,14,18)(10,16,15,12)(13,19,22,24)(17,23,21,20)(25,27,30,33)(26,29,31,39)(28,37,36,34)(32,40,35,38);(1,4,6,3)(2,5,7,8)(9,12,14,16)(10,11,15,18)(13,20,22,23)(17,19,21,24)(25,28,30,36)(26,32,31,35)(27,34,33,37)(29,38,39,40);(1,5)(2,3)(4,7)(6,8)(9,13,14,22)(10,17,15,21)(11,19,18,24)(12,20,16,23)(25,29)(26,33)(27,31)(28,38)(30,39)(32,37)(34,35)(36,40)Polycyclic presentation:
2^9|a^2=gh,d^2=g,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, 8841919);