Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12093 |
| automorphism | Aut(G) order 4096 |
| 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 | C24 | C2×D4 |
| LCS | G ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23 ◁ C2×C22⋊C4 ◁ G | |
| LECS | G ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
194 conjugacy classes of subgroups; only the 37 normal subgroups are shown.
This group has 194 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) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 40 points:
(1,2,4,7)(3,5,8,6)(9,10)(11,19)(12,16)(13,17)(14,18)(15,21)(20,36)(22,35)(23,34)(24,30)(25,31)(26,33)(27,37)(28,38)(29,39)(32,40);(2,5)(3,8)(6,7)(9,11,14,22)(10,15,18,28)(12,23,25,20)(13,21,26,38)(16,29,31,27)(17,19,33,35)(24,39,40,37)(30,34,32,36);(9,12,14,25)(10,16,18,31)(11,20,22,23)(13,24,26,40)(15,27,28,29)(17,32,33,30)(19,34,35,36)(21,37,38,39)Polycyclic presentation:
2^8|a^2=f,b^2=h,c^2=h,d^2=f,b^a=bd,c^a=ce,d^a=df,g^a=gh,c^b=ch,d^b=df,e^b=egh,d^c=dg,e^d=eh25 conjugacy classes — view the full character table.
| maximal subgroup of | #151199, #151345, C2×#12207, … |
| maximal normal subgroup of | #151199, #151345, C2×#12207, … |
| maximal quotient of | #125883, #125938, #126043, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 5195);