Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12099 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | B(4,𝔽2) |
| commutator, Frattini, cosocle | C2×C4⋊C4 | C23 |
| prime radical | C23 | #120 |
| LCS | G ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ G | |
| LECS | G ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
190 conjugacy classes of subgroups; only the 37 normal subgroups are shown.
This group has 190 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) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2,6,12)(3,15,18,28)(4,20,22,31)(5,24,16,10)(7,14,27,13)(8,23,30,17)(9,26,29,21)(11,25,32,19);(1,3,6,18)(2,8,12,30)(4,21,22,26)(5,23,16,17)(7,19,27,25)(9,32,29,11)(10,28,24,15)(13,20,14,31);(1,4,7,23)(2,9,14,10)(3,16,19,21)(5,25,26,18)(6,22,27,17)(8,28,31,32)(11,30,15,20)(12,29,13,24)Polycyclic presentation:
2^8|a^2=f,b^2=f,c^2=h,d^2=f,e^2=h,b^a=bd,c^a=ce,d^a=df,e^a=eh,g^a=gh,c^b=ch,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 | #151387, #151388, #151408, … |
| maximal normal subgroup of | #151387, #151388, #151408, … |
| maximal quotient of | #125973, #126013, #126080, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 5212);