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 | |
4615 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,4,6)(3,7,8,5)(9,10,14,15)(11,17,12,16)(13,21,22,18)(19,23,20,24)(25,26)(27,29)(28,32)(30,33)(31,34)(35,37)(36,39)(38,40);(1,2,4,6)(3,5,8,7)(10,15)(16,17)(18,21)(23,24)(25,27,31,28)(26,32,34,29)(30,35,38,36)(33,39,40,37);(1,2)(3,5)(4,6)(7,8)(9,11,14,12)(10,16,15,17)(13,19,22,20)(18,23,21,24)(25,27,31,28)(26,29,34,32)(30,35,38,36)(33,37,40,39);(3,8)(5,7)(9,12,14,11)(10,17,15,16)(13,19,22,20)(18,23,21,24)(25,28,31,27)(26,29,34,32)(30,35,38,36)(33,39,40,37);(1,2)(3,5)(4,6)(7,8)(9,10)(11,16)(12,17)(13,18)(14,15)(19,23)(20,24)(21,22)(25,29)(26,28)(27,34)(30,37)(31,32)(33,36)(35,40)(38,39);(1,3)(2,5)(4,8)(6,7)(9,13,14,22)(10,18,15,21)(11,19,12,20)(16,23,17,24)(25,30,31,38)(26,33,34,40)(27,35,28,36)(29,37,32,39)Polycyclic presentation:
2^9|a^2=h,b^2=gh,c^2=g,d^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=egi,e^d=eghi,f^d=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10315297);