Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12148 |
| automorphism | Aut(G) = C25×B(4,𝔽2) (order 2048) · Inn(G) = B(4,𝔽2) · Out(G) = C25 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | B(4,𝔽2) |
| commutator, Frattini, cosocle | C2×C22⋊C4 | C23 |
| prime radical | #392 | C22 |
| LCS | G ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23 ◁ C2×C22⋊C4 ◁ G | |
| LECS | G ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
232 conjugacy classes of subgroups; only the 37 normal subgroups are shown.
This group has 232 conjugacy classes of subgroups — too many to draw. Only its 37 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2,4,7)(3,5,8,6)(9,10,14,18)(11,19,21,24)(12,17,13,16)(15,22,23,20);(2,5)(3,8)(6,7)(9,11)(10,15)(12,20)(13,22)(14,21)(16,19)(17,24)(18,23);(9,12)(10,16,18,17)(11,20)(13,14)(15,19,23,24)(21,22)Polycyclic presentation:
2^8|a^2=f,c^2=gh,d^2=f,b^a=bd,c^a=ce,d^a=df,g^a=gh,d^b=df,e^b=egh,d^c=dg,e^c=eh,e^d=eh28 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#12243, #484664, #484665, … |
| maximal normal subgroup of | C2×#12243, #484664, #484665, … |
| maximal quotient of | #125920, #125925, #125939, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 5231);