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 | #2819 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4107 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,7)(3,6,8,4)(9,10,13,16)(11,14,17,21)(12,19,20,15)(18,24,23,22)(25,26)(27,35)(28,29)(30,32)(31,34)(33,40)(36,37)(38,39);(1,2)(3,4)(5,7)(6,8)(11,17)(14,21)(18,23)(22,24)(25,27,31,30)(26,32,34,35)(28,38,33,36)(29,37,40,39);(1,2)(3,4)(5,7)(6,8)(9,10)(11,14)(12,15)(13,16)(17,21)(18,22)(19,20)(23,24)(25,28,31,33)(26,29,34,40)(27,36,30,38)(32,39,35,37);(1,3,5,8)(2,6,7,4)(9,11,13,17)(10,14,16,21)(12,18,20,23)(15,22,19,24)(25,29)(26,33)(27,37)(28,34)(30,39)(31,40)(32,36)(35,38);(1,2)(3,6)(4,8)(5,7)(9,12,13,20)(10,15,16,19)(11,18,17,23)(14,22,21,24)(25,30,31,27)(26,32,34,35)(28,38,33,36)(29,39,40,37);(1,4,5,6)(2,3,7,8)Polycyclic presentation:
2^9|a^2=h,b^2=i,c^2=i,d^2=h,e^2=gh,f^2=gi,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, 8656903);