Finite Group DatabasebetaFinite Group Databasebeta| order | 1000 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C5 |
No aliases for this group.
| Sylow | C53 ⋊ Q8 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | D5×C5⋊D5 |
| commutator, Fitting, socle, prime radical, cosocle | C2×C53 | C22 |
| hypocenter | C53 | Q8 |
| LCS | G ▷ C2×C53 ▷ C53 | |
| UCS | C1 ◁ C2 | |
168 conjugacy classes of subgroups.
| transitive degree | μt(G) = 200 |
Polycyclic presentation:
2^3,5^3|a^2=c,b^2=c,b^a=bc,f^a=f^4,d^b=d^4,e^b=e^4109 conjugacy classes — view the full character table.
| maximal subgroup of | D5×#6039, #14522772, #14522775, … |
| maximal normal subgroup of | D5×#6039, #14522772, #14522775, … |
| maximal quotient of | #14522595, #14522596, #14522597, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1000, 140);