Finite Group DatabasebetaFinite Group Databasebeta| order | 1000 |
| automorphism | ? |
| homology | Abelianization (H1) = C8 · Schur multiplier (H2) = C1 |
No aliases for this group.
| derived, Sylow | H(3,𝔽5) ⋊ C8 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, hypercenter | C2×C5 | #1044 |
| commutator, hypocenter | H(3,𝔽5) | C8 |
| Fitting, prime radical | C2×H(3,𝔽5) | C4 |
| cosocle | #92525 | C2 |
| LCS | G ▷ H(3,𝔽5) | |
| UCS | C1 ◁ C2×C5 | |
40 conjugacy classes of subgroups.
| transitive degree | μt(G) = 200 |
Polycyclic presentation:
2^3,5^3|a^2=b,b^2=c,d^a=d^3,e^a=e^2,d^b=d^4,e^b=e^4,e^d=ef52 conjugacy classes — view the full character table.
| maximal subgroup of | #14522300, #14522301, #14522305, … |
| maximal normal subgroup of | #14522300, #14522301, #14522305, … |
| maximal quotient of | #14522109, C2×#11758649 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1000, 34);