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 | |
4519 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,7)(3,6,8,4)(9,10)(11,20)(12,14)(13,16)(15,17)(18,19)(21,24)(22,23)(25,26)(27,30)(28,35)(29,32)(31,36)(33,37)(34,40)(38,39);(1,3)(2,6)(4,7)(5,8)(9,11,15,13)(10,16,17,20)(12,21,19,23)(14,22,18,24)(27,33)(30,37)(34,39)(38,40);(1,2)(3,4)(5,7)(6,8)(10,17)(12,19)(16,20)(21,23)(25,26,29,32)(27,30,33,37)(28,31,36,35)(34,38,39,40);(1,3)(2,6)(4,7)(5,8)(9,12,15,19)(10,18,17,14)(11,21,13,23)(16,24,20,22)(25,26)(27,30)(28,31)(29,32)(33,37)(34,38)(35,36)(39,40);(1,4,5,6)(2,3,7,8)(9,13,15,11)(10,16,17,20)(12,23,19,21)(14,22,18,24)(25,27)(26,30)(28,34)(29,33)(31,38)(32,37)(35,40)(36,39);(1,3)(2,4)(5,8)(6,7)(9,14,15,18)(10,19,17,12)(11,22,13,24)(16,23,20,21)(25,28)(26,31)(27,34)(29,36)(30,38)(32,35)(33,39)(37,40)Polycyclic presentation:
2^9|a^2=i,b^2=g,c^2=ghi,d^2=g,e^2=gi,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, 9757880);