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 | |
4363 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10,15,17)(11,12,20,16)(13,18,24,22)(14,21,23,19)(25,26)(27,33)(28,32)(29,38)(30,34)(31,37)(35,40)(36,39);(1,2)(3,6)(4,7)(5,8)(9,11,15,20)(10,16,17,12)(13,23,24,14)(18,21,22,19)(25,27)(26,31)(28,35)(29,36)(30,37)(32,39)(33,34)(38,40);(1,3)(2,5)(4,8)(6,7)(10,17)(12,16)(18,22)(19,21)(25,28,30,29)(26,32,34,38)(27,35,37,36)(31,39,33,40);(1,2)(3,5)(4,7)(6,8)(9,12)(10,11)(13,19)(14,22)(15,16)(17,20)(18,23)(21,24)(25,29,30,28)(26,32,34,38)(27,36,37,35)(31,39,33,40);(1,3)(2,6)(4,8)(5,7)(9,13,15,24)(10,18,17,22)(11,14,20,23)(12,21,16,19)(25,26)(27,31)(28,32)(29,38)(30,34)(33,37)(35,39)(36,40);(1,2)(3,5)(4,7)(6,8)(9,14)(10,19)(11,13)(12,22)(15,23)(16,18)(17,21)(20,24)(25,27)(26,33)(28,35)(29,36)(30,37)(31,34)(32,40)(38,39)Polycyclic presentation:
2^9|a^2=h,b^2=h,c^2=gh,d^2=gh,e^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^d=egi,f^e=fgi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9995564);