Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2202 |
| 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 | C2×2−1+4 | C22 |
| LCS | G ▷ C2×Q8 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23 ◁ C22×Q8 ◁ G | |
| LECS | G ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1 | |
316 conjugacy classes of subgroups; only the 59 normal subgroups are shown.
This group has 316 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) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(2,4)(5,6,8,16,11,17,9,14)(7,18,21,12,22,29,19,30)(10,15,24,13,26,31,23,32)(20,33,34,36,25,27,35,28);(1,2,3,4)(5,7)(6,12)(8,19)(9,21)(10,25)(11,22)(13,27)(14,29)(15,28)(16,18)(17,30)(20,26)(23,35)(24,34)(31,36)(32,33);(5,8,11,9)(6,13,17,32)(7,19,22,21)(10,24,26,23)(12,27,30,33)(14,31,16,15)(18,28,29,36)(20,35,25,34)Polycyclic presentation:
2^8|a^2=f,b^2=df,c^2=h,d^2=h,e^2=h,f^2=h,b^a=bd,c^a=ce,e^a=egh,g^a=gh,d^b=dh,e^b=egh,f^b=fh,d^c=dg,e^c=eh,f^c=fg,e^d=eh,f^e=fh34 conjugacy classes — view the full character table.
| maximal subgroup of | #152196, #153306, C2×#13056, … |
| maximal normal subgroup of | #152196, #153306, C2×#13056, … |
| maximal quotient of | #136826, #136856, #137074, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6044);