Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24468 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#120 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C2×AGL(1,ℤ/8) | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
525 conjugacy classes of subgroups; only the 123 normal subgroups are shown.
This group has 525 conjugacy classes of subgroups — too many to draw. Only its 123 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 — 4 generators on 40 points:
(1,2)(3,7)(4,5)(6,8)(9,10,14,18)(11,15,20,29)(12,16,23,32)(13,25,27,17)(19,35,36,28)(21,34,26,30)(22,31,37,40)(24,39,38,33);(2,5)(6,7)(9,11)(10,15)(12,22)(13,26)(14,20)(16,31)(17,34)(18,29)(19,24)(21,27)(23,37)(25,30)(28,33)(32,40)(35,39)(36,38);(9,12)(10,16)(11,19,20,36)(13,24)(14,23)(15,28,29,35)(17,33)(18,32)(21,22,26,37)(25,39)(27,38)(30,31,34,40);(1,3,4,8)(2,6,5,7)(12,23)(16,32)(19,36)(22,37)(24,38)(28,35)(31,40)(33,39)Polycyclic presentation:
2^8|a^2=h,c^2=f,d^2=e,g^2=h,b^a=be,c^a=cf,d^a=de,g^a=gh,c^b=cg,f^b=fh,g^b=gh,d^c=dh40 conjugacy classes — view the full character table.
| maximal subgroup of | #250396, #250398, #250404, … |
| maximal normal subgroup of | #250396, #250398, #250404, … |
| maximal quotient of | #293332, #293335, #293340, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 18582);