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 | |
4417 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,8)(3,4,7,6)(12,20)(16,21)(18,24)(22,23)(25,26)(27,28)(29,34)(30,35)(31,32)(33,40)(36,38)(37,39);(3,7)(4,6)(9,10)(11,17)(12,16)(13,14)(15,19)(18,23)(20,21)(22,24)(26,32)(27,33)(34,35)(36,37);(1,3,5,7)(2,6,8,4)(10,14)(15,17)(16,21)(22,23)(25,27,31,33)(26,28,32,40)(29,36,30,37)(34,38,35,39);(1,4,5,6)(2,7,8,3)(9,10)(11,15)(12,16)(13,14)(17,19)(18,22)(20,21)(23,24)(25,28)(26,33)(27,32)(29,38)(30,39)(31,40)(34,37)(35,36);(9,11,13,19)(10,15,14,17)(12,18,20,24)(16,22,21,23)(25,29,31,30)(26,34,32,35)(27,36,33,37)(28,38,40,39);(9,12)(10,16)(11,18)(13,20)(14,21)(15,22)(17,23)(19,24)(25,30,31,29)(26,35,32,34)(27,37,33,36)(28,39,40,38)Polycyclic presentation:
2^9|a^2=h,c^2=gh,d^2=h,e^2=hi,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fghi,c^b=ci,d^b=dh,e^b=eghi,d^c=dghi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8000798);