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 | |
4675 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,4)(6,8)(9,10)(11,18)(12,20)(13,15)(14,16)(17,21)(19,23)(22,24)(25,26,31,35)(27,32,36,40)(28,30,34,39)(29,38,37,33);(3,7)(6,8)(9,11,15,14)(10,16,13,18)(12,19,21,22)(17,23,20,24)(27,36)(29,37)(32,40)(33,38);(1,2,5,4)(3,6,7,8)(25,27,31,36)(26,32,35,40)(28,37,34,29)(30,33,39,38);(1,3,5,7)(2,6,4,8)(9,12)(10,17)(11,19)(13,20)(14,22)(15,21)(16,23)(18,24)(25,28)(26,30)(27,37)(29,36)(31,34)(32,33)(35,39)(38,40);(1,4,5,2)(3,8,7,6)(9,13,15,10)(11,16,14,18)(12,20,21,17)(19,23,22,24)(25,29,31,37)(26,33,35,38)(27,28,36,34)(30,32,39,40);(9,14,15,11)(10,16,13,18)(12,19,21,22)(17,24,20,23)(25,30)(26,34)(27,33)(28,35)(29,32)(31,39)(36,38)(37,40)Polycyclic presentation:
2^9|a^2=i,b^2=g,c^2=hi,d^2=h,e^2=ghi,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,e^b=egi,f^d=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8400445);