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:
(1,2,5,3)(4,7,8,6)(9,10,14,16)(11,15,18,12)(13,21,23,17)(19,20,24,22)(25,26,32,36)(27,37,39,33)(28,34,29,31)(30,35,38,40);(2,3)(6,7)(9,11)(10,15)(12,16)(13,19)(14,18)(17,22)(20,21)(23,24)(25,27)(26,33)(28,30)(29,38)(31,35)(32,39)(34,40)(36,37);(1,3,5,2)(4,7,8,6)(10,16)(11,18)(17,21)(19,24)(25,28,32,29)(26,34,36,31)(27,30,39,38)(33,40,37,35);(1,4,5,8)(2,6,3,7)(9,11)(10,12)(13,19)(14,18)(15,16)(17,20)(21,22)(23,24)(25,29,32,28)(26,31,36,34)(27,38,39,30)(33,35,37,40);(9,12)(10,11)(13,22)(14,15)(16,18)(17,24)(19,21)(20,23)(25,30,32,38)(26,35,36,40)(27,28,39,29)(31,37,34,33);(9,13,14,23)(10,17,16,21)(11,19,18,24)(12,20,15,22)(25,31)(26,28)(27,35)(29,36)(30,33)(32,34)(37,38)(39,40)Polycyclic presentation:
2^9|a^2=gh,c^2=g,d^2=g,e^2=ghi,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,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, 9623314);