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 | |
4536 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10,15,17)(11,14,12,16)(13,18,22,24)(19,21,20,23)(25,26,30,28)(27,31,35,33)(29,36,37,32)(34,39,40,38);(1,2,5,8)(3,4,7,6)(9,11,15,12)(10,16,17,14)(13,20,22,19)(18,21,24,23)(25,27,30,35)(26,31,28,33)(29,34,37,40)(32,38,36,39);(1,3)(2,6)(4,8)(5,7)(10,17)(11,12)(18,24)(19,20)(26,28)(31,33)(32,36)(38,39);(1,4)(2,7)(3,8)(5,6)(9,12,15,11)(10,14,17,16)(13,20,22,19)(18,21,24,23)(25,28,30,26)(27,33,35,31)(29,36,37,32)(34,39,40,38);(1,3)(2,4)(5,7)(6,8)(9,13,15,22)(10,18,17,24)(11,19,12,20)(14,21,16,23)(25,27)(26,31)(28,33)(29,34)(30,35)(32,38)(36,39)(37,40);(1,4)(2,7)(3,8)(5,6)(9,14)(10,11)(12,17)(13,21)(15,16)(18,19)(20,24)(22,23)(25,29,30,37)(26,32,28,36)(27,34,35,40)(31,38,33,39)Polycyclic presentation:
2^9|a^2=h,b^2=ghi,d^2=h,e^2=g,f^2=gh,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, 10004342);