Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2196 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | B(4,𝔽2) |
| commutator | C2×D4 | C2×C42 |
| Frattini, cosocle | C22×#41 | C23 |
| socle | C22 | #1790 |
| prime radical | C2×#532 | C22 |
| LCS | G ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C4 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C22×#41 ▷ C22 ▷ C2 ▷ C1 | |
1141 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 20 points:
(5,6)(7,15,18,19)(8,12,17,11)(9,13)(10,14)(16,20);(1,2,3,4)(6,11)(12,14)(13,15)(19,20);(5,7)(6,12)(8,9)(10,18)(11,14)(13,19)(15,20)(16,17)Polycyclic presentation:
2^9|a^2=f,b^2=g,e^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.
| maximal subgroup of | #72282075, #72282124, #72282140, … |
| maximal normal subgroup of | #72282075, #72282124, #72282140, … |
| maximal quotient of | #72282471, #72282675 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 53831);