Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2204 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | B(4,𝔽2) |
| commutator | C2×Q8 | C22×C4 |
| Frattini, cosocle | C22×Q8 | C23 |
| prime radical | C2×#454 | C2 |
| LCS | G ▷ C2×Q8 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23 ◁ C22×Q8 ◁ G | |
| LECS | G ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1 | |
350 conjugacy classes of subgroups; only the 59 normal subgroups are shown.
This group has 350 conjugacy classes of subgroups — too many to draw. Only its 59 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 20 points:
(2,4)(5,6,7,11,10,15,8,13)(9,18,17,20,19,14,16,12);(1,2,3,4)(6,11)(7,8)(12,14)(13,15)(16,17)(18,20);(5,7)(6,12)(8,10)(9,17)(11,18)(13,14)(15,20)(16,19)Polycyclic presentation:
2^8|a^2=f,b^2=df,d^2=h,e^2=h,f^2=h,b^a=bd,c^a=ce,e^a=egh,g^a=gh,c^b=ch,d^b=dh,e^b=egh,f^b=fh,d^c=dg,e^c=eh,f^c=fg,e^d=eh,f^e=fh40 conjugacy classes — view the full character table.
| maximal subgroup of | #152179, #153308, C2×#13057, … |
| maximal normal subgroup of | #152179, #153308, C2×#13057, … |
| maximal quotient of | #136827, #136853, #137075, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6045);