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 | #59142 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4501 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,6)(3,7)(9,10)(11,14)(12,16)(13,17)(15,18)(19,22)(20,21)(23,24)(25,26,31,34)(27,29,36,40)(28,33,30,32)(35,38,37,39);(1,2,5,6)(3,4,7,8)(9,11)(10,14)(12,19)(13,18)(15,17)(16,22)(20,23)(21,24)(25,27)(26,29)(28,37)(30,35)(31,36)(32,38)(33,39)(34,40);(1,2,5,6)(3,4,7,8)(11,18)(14,15)(19,24)(22,23)(25,28,31,30)(26,32,34,33)(27,35,36,37)(29,39,40,38);(2,6)(4,8)(9,11)(10,15)(12,19)(13,18)(14,17)(16,23)(20,22)(21,24)(25,29,31,40)(26,27,34,36)(28,38,30,39)(32,37,33,35);(1,3)(2,4)(5,7)(6,8)(9,12)(10,16)(11,19)(13,21)(14,22)(15,23)(17,20)(18,24)(25,30,31,28)(26,33,34,32)(27,35,36,37)(29,39,40,38);(1,4)(2,7)(3,6)(5,8)(9,10)(11,14)(12,20)(13,17)(15,18)(16,21)(19,23)(22,24)(25,27)(26,29)(28,35)(30,37)(31,36)(32,39)(33,38)(34,40)Polycyclic presentation:
2^9|a^2=gh,b^2=g,c^2=h,d^2=gh,e^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, 8635703);