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 | |
4443 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,4,5)(3,7,8,6)(9,10)(11,17)(12,16)(13,14)(15,19)(18,24)(20,22)(21,23)(25,26)(27,35)(28,32)(29,33)(30,34)(31,37)(36,40)(38,39);(2,5)(6,7)(10,14)(15,17)(16,20)(23,24)(25,27,30,37)(26,31,34,35)(28,29,36,39)(32,38,40,33);(1,2)(3,6)(4,5)(7,8)(10,14)(15,17)(16,20)(23,24)(25,28,30,36)(26,32,34,40)(27,29,37,39)(31,38,35,33);(1,2)(3,6)(4,5)(7,8)(9,10)(11,15)(12,20)(13,14)(16,22)(17,19)(18,24)(21,23)(25,28)(26,32)(27,29)(30,36)(31,38)(33,35)(34,40)(37,39);(1,3)(2,6)(4,8)(5,7)(9,11)(10,15)(12,21)(13,19)(14,17)(16,24)(18,22)(20,23)(25,26)(27,31)(28,32)(29,38)(30,34)(33,39)(35,37)(36,40);(9,12)(10,16)(11,18)(13,22)(14,20)(15,23)(17,24)(19,21)(25,29)(26,33)(27,36)(28,37)(30,39)(31,32)(34,38)(35,40)Polycyclic presentation:
2^9|a^2=i,b^2=gh,c^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, 8638069);