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 |
| prime radical | #52543 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4176 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,6,7)(3,5,4,8)(9,10)(11,19)(12,22)(13,18)(14,16)(15,21)(17,20)(23,24)(25,26,31,30)(27,32,37,36)(28,38,33,29)(34,40,39,35);(2,7)(3,4)(9,11,14,21)(10,15,16,19)(12,13,20,24)(17,18,22,23)(25,27,31,37)(26,32,30,36)(28,34,33,39)(29,35,38,40);(1,3,6,4)(2,8,7,5)(10,16)(15,19)(17,22)(18,23)(25,28)(26,29)(27,34)(30,38)(31,33)(32,35)(36,40)(37,39);(1,4,6,3)(2,5,7,8)(9,10)(11,19)(12,17)(13,23)(14,16)(15,21)(18,24)(20,22)(25,27)(26,32)(28,34)(29,35)(30,36)(31,37)(33,39)(38,40);(1,5)(2,3)(4,7)(6,8)(9,12)(10,17)(11,13)(14,20)(15,18)(16,22)(19,23)(21,24)(25,29,31,38)(26,33,30,28)(27,35,37,40)(32,39,36,34);(9,13)(10,18)(11,20)(12,21)(14,24)(15,22)(16,23)(17,19)(25,30,31,26)(27,36,37,32)(28,38,33,29)(34,40,39,35)Polycyclic presentation:
2^9|a^2=ghi,b^2=hi,c^2=i,d^2=i,e^2=gh,f^2=gh,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8653213);