Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D4×AGL(1,ℤ/8) |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C23×D4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| socle | C22 | C22×C4×D4 |
| prime radical | C2×#135 | C23 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C8×Q8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1693 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 40 points:
(2,6)(7,8)(9,10,14,18,15,19,13,28)(11,20,23,16,21,32,29,30)(12,24,26,39,27,17,25,33)(22,36,37,38,35,34,40,31);(1,2,5,6)(3,8,4,7)(11,21)(16,30)(20,32)(22,35)(23,29)(31,38)(34,36)(37,40);(9,11)(10,16)(12,22)(13,29)(14,23)(15,21)(17,31)(18,32)(19,30)(20,28)(24,38)(25,40)(26,37)(27,35)(33,36)(34,39);(1,3,5,4)(2,7,6,8);(1,4,5,3)(2,7,6,8)(9,12,14,26,15,27,13,25)(10,17,18,33,19,24,28,39)(11,22,23,37,21,35,29,40)(16,31,32,36,30,38,20,34)Polycyclic presentation:
2^9|a^2=h,b^2=f,d^2=f,e^2=fh,g^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ei,g^a=gi,c^b=ci,d^b=df,g^c=gi110 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6292341);