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 | |
4490 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10,14,16)(11,18,21,15)(12,20,22,17)(13,23,24,19)(25,26,32,36)(27,31,29,34)(28,38,37,33)(30,35,39,40);(1,2,5,8)(3,4,7,6)(9,11,14,21)(10,15,16,18)(12,19,22,23)(13,20,24,17)(25,27,32,29)(26,31,36,34)(28,39,37,30)(33,35,38,40);(1,3)(2,6)(4,8)(5,7)(10,16)(12,22)(15,18)(19,23)(25,28,32,37)(26,33,36,38)(27,30,29,39)(31,40,34,35);(1,4)(2,7)(3,8)(5,6)(9,12,14,22)(10,17,16,20)(11,19,21,23)(13,18,24,15)(25,29,32,27)(26,34,36,31)(28,30,37,39)(33,40,38,35);(1,3)(2,4)(5,7)(6,8)(9,11)(10,18)(12,23)(13,17)(14,21)(15,16)(19,22)(20,24)(25,30)(26,35)(27,37)(28,29)(31,33)(32,39)(34,38)(36,40);(1,4)(2,7)(3,8)(5,6)(9,13)(10,19)(11,20)(12,18)(14,24)(15,22)(16,23)(17,21)(25,31,32,34)(26,27,36,29)(28,40,37,35)(30,38,39,33)Polycyclic presentation:
2^9|a^2=h,b^2=gi,c^2=gh,d^2=h,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, 10004452);