Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #360 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C8 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C42 | #102 |
| commutator | C8 | C2×C4×C8 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C22 | C2×#340 |
| prime radical | C24 | #103 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C42 ◁ C22×C42 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
548 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 40 points:
(1,2,4,7,5,8,3,6)(9,10,13,18,15,21,32,20)(11,22,25,36,23,37,27,40)(12,28,30,17,31,19,14,33)(16,26,35,24,34,39,38,29);(2,6,8,7)(9,11,14,26,32,25,31,24,15,23,30,39,13,27,12,29)(10,16,19,36,18,38,33,22,21,34,28,40,20,35,17,37);(11,23)(16,34)(22,37)(24,29)(25,27)(26,39)(35,38)(36,40)Polycyclic presentation:
2^9|a^2=e,b^2=f,d^2=g,e^2=h,f^2=gh,g^2=i,b^a=bd,d^a=dg,f^a=fgi,g^a=gi,c^b=ci,d^b=di,e^b=ei128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 33979);