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 | #2408 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4422 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)(9,10,15,19)(11,20,22,16)(12,13,21,17)(14,18,24,23)(25,26,31,35)(27,32,36,40)(28,30,34,39)(29,38,37,33);(3,7)(6,8)(9,11,15,22)(10,16,19,20)(12,14,21,24)(13,23,17,18)(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)(11,14)(13,19)(15,21)(16,18)(20,23)(22,24)(25,28)(26,30)(27,37)(29,36)(31,34)(32,33)(35,39)(38,40);(1,4,5,2)(3,8,7,6)(9,13,15,17)(10,12,19,21)(11,18,22,23)(14,16,24,20)(25,29,31,37)(26,33,35,38)(27,28,36,34)(30,32,39,40);(9,14)(10,18)(11,21)(12,22)(13,16)(15,24)(17,20)(19,23)(25,30)(26,34)(27,33)(28,35)(29,32)(31,39)(36,38)(37,40)Polycyclic presentation:
2^9|a^2=gi,b^2=g,c^2=hi,d^2=h,e^2=ghi,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, 8400431);