Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #95118 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C22×C4 |
| unambiguous semidirect product | C4⋊#1308 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#95 |
| commutator, prime radical | C24 | C2×C42 |
| Frattini, cosocle | C22×C42 | C23 |
| LCS | G ▷ C24 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C43 ◁ G | |
| LECS | G ▷ C22×C42 ▷ C22 ▷ C1 | |
592 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 3 generators on 36 points:
(2,4)(5,6,8,13,10,12,9,14)(7,15,16,20,17,11,18,19)(21,22,24,27,25,28,32,35)(23,29,31,36,30,26,33,34);(5,7,9,16,10,17,8,18)(6,11,14,19,12,15,13,20)(21,23)(22,26,28,29)(24,31)(25,30)(27,34,35,36)(32,33);(1,2,3,4)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=e,f^2=hi,g^2=h,b^a=bd,c^a=ce,d^a=dh,g^a=gi,d^b=di,f^b=fh80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 16130);