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 | |
4566 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)(3,7)(4,6)(5,8)(9,10,14,16)(11,17,18,12)(13,15,21,22)(19,23,24,20)(25,26,31,35)(27,36,39,32)(28,30,38,33)(29,34,37,40);(1,2,4,6)(3,5,8,7)(9,10)(11,17)(12,18)(13,15)(14,16)(19,23)(20,24)(21,22)(25,27)(26,32)(28,37)(29,38)(30,34)(31,39)(33,40)(35,36);(1,2)(3,5)(4,6)(7,8)(11,18)(12,17)(13,21)(15,22)(25,28,31,38)(26,33,35,30)(27,37,39,29)(32,40,36,34);(1,3,4,8)(2,5,6,7)(9,11,14,18)(10,12,16,17)(13,19,21,24)(15,20,22,23)(25,28)(26,33)(27,29)(30,35)(31,38)(32,34)(36,40)(37,39);(9,12)(10,11)(13,20)(14,17)(15,19)(16,18)(21,23)(22,24)(25,29)(26,34)(27,38)(28,39)(30,32)(31,37)(33,36)(35,40);(9,13)(10,15)(11,19)(12,20)(14,21)(16,22)(17,23)(18,24)(25,30,31,33)(26,28,35,38)(27,34,39,40)(29,32,37,36)Polycyclic presentation:
2^9|a^2=gi,b^2=i,c^2=h,d^2=gh,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,d^c=dgi,e^c=egi,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, 8969353);