Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24802 |
| automorphism | Aut(G) order 65536 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#121 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C23 | C2×#41 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
363 conjugacy classes of subgroups; only the 123 normal subgroups are shown.
This group has 363 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) = 128 |
A minimal faithful permutation representation — 4 generators on 40 points:
(1,2,5,8)(3,7,4,6)(9,10)(11,16)(12,24)(13,28)(14,18)(15,19)(17,26)(20,36)(21,38)(22,31)(23,32)(25,30)(27,34)(29,35)(33,37)(39,40);(1,3,5,4)(2,6,8,7)(9,11,14,22,15,23,13,21)(10,16,18,31,19,32,28,38)(12,25,26,37,27,39,29,20)(17,33,34,40,35,36,24,30);(1,3,5,4)(2,7,8,6)(9,12,13,26,15,27,14,29)(10,17,18,24,19,35,28,34)(11,20,21,25,23,37,22,39)(16,30,31,36,32,40,38,33);(1,4,5,3)(2,7,8,6)Polycyclic presentation:
2^8|a^2=e,b^2=eg,c^2=ef,d^2=e,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=de,f^a=fh,c^b=cg,g^c=gh46 conjugacy classes — view the full character table.
| maximal subgroup of | #139416, #139419, #143242, … |
| maximal normal subgroup of | #139416, #139419, #143242, … |
| maximal quotient of | #325557, #325563, #325569, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 21750);