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 | |
4373 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,7)(3,4,8,6)(9,10)(11,17)(12,15)(13,16)(14,19)(18,23)(20,21)(22,24)(25,26,31,35)(27,32,37,29)(28,38,36,33)(30,40,39,34);(1,3,5,8)(2,6,7,4)(9,10,13,16)(11,14,19,17)(12,15,21,20)(18,22,24,23)(25,27)(26,32)(28,39)(29,35)(30,36)(31,37)(33,40)(34,38);(1,3)(2,4)(5,8)(6,7)(25,28,31,36)(26,33,35,38)(27,30,37,39)(29,40,32,34);(1,4)(2,3)(5,6)(7,8)(9,11)(10,14)(12,18)(13,19)(15,22)(16,17)(20,23)(21,24)(25,29,31,32)(26,27,35,37)(28,34,36,40)(30,33,39,38);(9,10)(11,14)(12,20)(13,16)(15,21)(17,19)(18,23)(22,24)(25,27)(26,32)(28,30)(29,35)(31,37)(33,34)(36,39)(38,40);(9,12)(10,15)(11,18)(13,21)(14,22)(16,20)(17,23)(19,24)(25,30)(26,34)(27,36)(28,37)(29,33)(31,39)(32,38)(35,40)Polycyclic presentation:
2^9|a^2=i,b^2=g,c^2=h,d^2=h,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, 9690499);