Finite Group DatabasebetaFinite Group Databasebeta| order | 1600 |
| automorphism | ? |
| homology | Abelianization (H1) = C42 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| Sylow | C52 ⋊ H(3,ℤ/4) |
| subgroup | G/H | |
|---|---|---|
| center | C2 | #11716795 |
| commutator | #1039 | C42 |
| Frattini, hypercenter | C23 | D52⋊C2 |
| Fitting, prime radical | C23×C52 | D4 |
| socle | C2×C52 | #95 |
| cosocle | C22×#1039 | C22 |
| hypocenter | C52 | H(3,ℤ/4) |
| LCS | G ▷ #1039 ▷ C52 | |
| UCS | C1 ◁ C2 ◁ C23 | |
204 conjugacy classes of subgroups; only the 33 normal subgroups are shown.
This group has 204 conjugacy classes of subgroups — too many to draw. Only its 33 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 80 |
Polycyclic presentation:
2^6,5^2|a^2=c,b^2=de,c^2=f,b^a=bcd,e^a=ef,g^a=g^2,h^a=h^3,c^b=cf,d^b=df,e^b=ef,g^b=h^4,h^b=g^4,g^c=g^4,h^c=h^470 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1600, 2357);