Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2204 |
| automorphism | Aut(G) = #72234841 (order 2048) · Inn(G) = B(4,𝔽2) · Out(G) = C22×D4 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | B(4,𝔽2) |
| commutator | C2×Q8 | C22×C4 |
| Frattini, cosocle | C2×#41 | C23 |
| socle | C2 | #1867 |
| prime radical | #532 | C22 |
| LCS | G ▷ C2×Q8 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C2×C4 ◁ C2×#41 ◁ G | |
| LECS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
208 conjugacy classes of subgroups; only the 57 normal subgroups are shown.
This group has 208 conjugacy classes of subgroups — too many to draw. Only its 57 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2,7,12,8,13,6,23)(3,14,18,25,19,31,17,29)(4,11,20,10,5,21,22,30)(9,24,27,15,28,32,26,16);(1,3,7,18,8,19,6,17)(2,9,12,27,13,28,23,26)(4,16,20,24,5,15,22,32)(10,29,21,14,30,25,11,31);(1,4)(2,10)(3,15)(5,8)(6,22)(7,20)(9,25)(11,23)(12,21)(13,30)(14,26)(16,19)(17,24)(18,32)(27,31)(28,29)Polycyclic presentation:
2^8|a^2=f,b^2=f,d^2=h,e^2=h,f^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=eh,g^a=gh,c^b=ch,d^b=dh,e^b=egh,d^c=dg,e^c=eh,e^d=eh40 conjugacy classes — view the full character table.
| maximal subgroup of | #152467, #152473, #152484, … |
| maximal normal subgroup of | #152467, #152473, #152484, … |
| maximal quotient of | #135944, #135945, #135962, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 5963);