Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of B(4,𝔽2) |
| automorphism | ? |
| homology | Abelianization (H1) = C43 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | #120 |
| commutator | C23 | C43 |
| Frattini, prime radical, cosocle | C26 | C23 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C26 ◁ G | |
| LECS | G ▷ C26 ▷ C2 ▷ C1 | |
3678 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 20 points:
(1,2)(3,4)(13,14)(15,19,20,17)(16,18);(5,6,7,8)(13,15)(14,17)(16,20)(18,19);(2,4)(9,10,11,12)(13,14)(15,17)(16,18)(19,20)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=h,b^a=bd,c^a=ce,d^a=di,e^b=ei,f^b=fi,d^c=di128 conjugacy classes — view the full character table.
| Fitting subgroup of | #72267335 |
| maximal subgroup of | C4×#10935, #72267335 |
| maximal normal subgroup of | C4×#10935, #72267335 |
| maximal quotient of | #72211321 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 30659);