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 | |
4196 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(10,11)(15,17)(16,18)(23,24)(25,26,31,33)(27,32,34,29)(28,30,37,40)(35,36,39,38);(1,2)(3,7)(4,5)(6,8)(9,10,14,11)(12,17,20,15)(13,16,22,18)(19,24,21,23)(25,27,31,34)(26,32,33,29)(28,35,37,39)(30,36,40,38);(1,3)(2,6)(4,7)(5,8)(26,33)(27,34)(28,37)(36,38);(9,11,14,10)(12,17,20,15)(13,18,22,16)(19,24,21,23)(25,28)(26,30)(27,35)(29,38)(31,37)(32,36)(33,40)(34,39);(1,4,5,2)(3,7,8,6)(9,12)(10,15)(11,17)(13,21)(14,20)(16,24)(18,23)(19,22)(25,29)(26,27)(28,38)(30,35)(31,32)(33,34)(36,37)(39,40);(1,3)(2,7)(4,6)(5,8)(9,13,14,22)(10,16,11,18)(12,19,20,21)(15,23,17,24)(25,30)(26,28)(27,36)(29,39)(31,40)(32,35)(33,37)(34,38)Polycyclic presentation:
2^9|a^2=h,b^2=gh,d^2=g,e^2=hi,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eghi,d^c=dh,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9744024);