Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33359 |
| automorphism | Aut(G) order 24576 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C22×#99 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C2×Q8 | C24 |
| prime radical | #3498 | C2 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×Q8 ◁ C2×H(5,𝔽2) ◁ G | |
| LECS | G ▷ C2×Q8 ▷ C22 ▷ C2 ▷ C1 | |
656 conjugacy classes of subgroups; only the 181 normal subgroups are shown.
This group has 656 conjugacy classes of subgroups — too many to draw. Only its 181 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 16 |
A minimal faithful permutation representation — 4 generators on 16 points:
(1,2,6,12,7,13,3,9)(4,10,15,8,5,11,14,16);(1,3,7,6)(2,8,13,16)(4,14,5,15)(9,11,12,10);(1,3,7,6)(2,9,13,12)(4,15,5,14)(8,11,16,10);(1,4,7,5)(2,10,13,11)(3,14,6,15)(8,9,16,12)Polycyclic presentation:
2^8|a^2=f,b^2=h,c^2=h,d^2=h,e^2=h,f^2=h,b^a=be,e^a=eg,g^a=gh,e^b=eh,f^b=fgh,d^c=dh,f^e=fh46 conjugacy classes — view the full character table.
| Fitting subgroup of | #33373⋊C3 |
| maximal subgroup of | #151832, #152707, #152743, … |
| maximal normal subgroup of | #151832, #152707, #152743, … |
| maximal quotient of | #480017, #480049, #480118, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 26361);