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 | |
4516 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)(3,8)(4,7)(5,6)(9,10,14,13)(11,17,18,15)(12,16,20,22)(19,24,21,23)(25,26,30,28)(27,31,33,35)(29,32,38,36)(34,39,37,40);(1,3,6,4)(2,7,5,8)(10,13)(11,18)(16,22)(19,21)(25,27)(26,31)(28,35)(29,37)(30,33)(32,40)(34,38)(36,39);(1,2,6,5)(3,7,4,8)(12,20)(16,22)(19,21)(23,24)(27,33)(31,35)(34,37)(39,40);(1,2,6,5)(3,8,4,7)(9,11,14,18)(10,15,13,17)(12,21,20,19)(16,24,22,23)(25,28,30,26)(27,31,33,35)(29,36,38,32)(34,39,37,40);(1,4,6,3)(2,7,5,8)(9,12,14,20)(10,16,13,22)(11,19,18,21)(15,23,17,24)(25,27)(26,31)(28,35)(29,34)(30,33)(32,39)(36,40)(37,38);(1,5,6,2)(3,7,4,8)(9,13,14,10)(11,17,18,15)(12,22,20,16)(19,24,21,23)(25,29)(26,32)(27,34)(28,36)(30,38)(31,39)(33,37)(35,40)Polycyclic presentation:
2^9|a^2=ghi,b^2=h,c^2=h,d^2=gi,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,f^b=fgi,e^c=egi,e^d=eghi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10259722);