Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2196 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | B(4,𝔽2) |
| commutator | C2×D4 | C2×C42 |
| Frattini, prime radical, cosocle | C23×D4 | C23 |
| LCS | G ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24 ◁ C23×D4 ◁ G | |
| LECS | G ▷ C23×D4 ▷ C22 ▷ C2 ▷ C1 | |
1304 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2)(3,4)(9,10,13,18,14,15,12,17)(11,19,22,16,20,24,21,23);(5,6,7,8)(10,15)(16,23);(2,4)(9,11,14,20)(10,16,15,23)(12,22,13,21)(17,24,18,19)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=i,e^2=i,f^2=i,b^a=bd,c^a=ce,d^a=dh,h^a=hi,e^b=eh,f^b=fh,d^c=dhi,e^c=ei,f^c=fi,e^d=ei,f^d=fi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 53836);