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 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4501 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)(3,5)(4,8)(6,7)(9,10)(11,13)(12,20)(14,15)(16,18)(17,22)(19,23)(21,24)(25,26,30,31)(27,28,35,32)(29,33,37,38)(34,36,40,39);(2,7)(3,8)(10,15)(13,16)(17,20)(21,23)(26,31)(28,32)(29,37)(34,40);(1,3,6,8)(2,4,7,5)(10,15)(11,18)(17,20)(19,24)(25,27,30,35)(26,28,31,32)(29,34,37,40)(33,36,38,39);(1,4,6,5)(2,3,7,8)(9,11,14,18)(10,16,15,13)(12,19,22,24)(17,23,20,21)(25,27)(26,32)(28,31)(29,34)(30,35)(33,39)(36,38)(37,40);(1,2)(3,4)(5,8)(6,7)(9,12,14,22)(10,17,15,20)(11,19,18,24)(13,21,16,23)(25,28,30,32)(26,27,31,35)(29,36,37,39)(33,34,38,40);(1,5,6,4)(2,8,7,3)(9,13,14,16)(10,18,15,11)(12,21,22,23)(17,24,20,19)(25,29,30,37)(26,33,31,38)(27,34,35,40)(28,36,32,39)Polycyclic presentation:
2^9|a^2=gh,c^2=ghi,d^2=h,e^2=gi,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eghi,f^b=fgi,d^c=dgi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8664226);