Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #327 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C8 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4×C8 | C22⋊C4 |
| commutator | C2×C4 | C2×C4×C8 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C22 | C2×#336 |
| prime radical | C24 | C4⋊C8 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C4×C8 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
469 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)(3,5)(4,7)(6,8)(9,10,12,18,15,19,25,33,14,17,13,26,24,32,27,37)(11,16,21,29,22,31,35,39,20,28,23,30,34,38,36,40);(2,5,7,6)(9,11,13,21,15,22,27,35,14,20,12,23,24,34,25,36)(10,16,18,30,19,31,33,40,17,28,26,29,32,38,37,39);(11,20)(16,28)(21,23)(22,34)(29,30)(31,38)(35,36)(39,40)Polycyclic presentation:
2^9|a^2=e,b^2=f,d^2=gi,e^2=h,f^2=h,h^2=i,b^a=bd,d^a=dg,f^a=fg,c^b=ci,d^b=di,e^b=ei152 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 33933);