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 | |
4617 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)(11,19)(12,21)(13,18)(14,15)(16,20)(17,22)(23,24)(25,26,31,35)(27,32,36,40)(28,30,34,39)(29,38,37,33);(2,5)(6,7)(10,15)(12,22)(16,19)(18,24)(27,36)(29,37)(32,40)(33,38);(1,2)(3,6)(4,5)(7,8)(25,27,31,36)(26,32,35,40)(28,37,34,29)(30,33,39,38);(1,2)(3,6)(4,5)(7,8)(9,11,14,20)(10,16,15,19)(12,18,22,24)(13,21,23,17)(25,28)(26,30)(27,37)(29,36)(31,34)(32,33)(35,39)(38,40);(1,3,4,8)(2,6,5,7)(9,12)(10,17)(11,18)(13,16)(14,22)(15,21)(19,23)(20,24)(25,29,31,37)(26,33,35,38)(27,28,36,34)(30,32,39,40);(9,13)(10,18)(11,17)(12,16)(14,23)(15,24)(19,22)(20,21)(25,30)(26,34)(27,33)(28,35)(29,32)(31,39)(36,38)(37,40)Polycyclic presentation:
2^9|a^2=i,c^2=hi,d^2=gh,e^2=i,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, 8400111);