Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24390 |
| automorphism | Aut(G) order 65536 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C22×#41 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
733 conjugacy classes of subgroups; only the 151 normal subgroups are shown.
This group has 733 conjugacy classes of subgroups — too many to draw. Only its 151 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:
(2,3)(9,10,14,18)(11,15,20,28)(12,22,25,37)(13,26,23,17)(16,30,31,24)(19,32,34,40)(21,36,33,29)(27,38,39,35);(1,2,4,3)(5,6)(7,8)(9,11)(10,15)(12,19)(13,21)(14,20)(16,27)(17,29)(18,28)(22,32)(23,33)(24,35)(25,34)(26,36)(30,38)(31,39)(37,40);(6,8)(9,12,23,30,14,25,13,24)(10,16,26,37,18,31,17,22)(11,19,33,38,20,34,21,35)(15,27,36,40,28,39,29,32);(1,3,4,2)(11,20)(15,28)(19,34)(21,33)(27,39)(29,36)(32,40)(35,38)Polycyclic presentation:
2^8|a^2=h,b^2=e,c^2=fh,d^2=e,f^2=h,b^a=be,c^a=cf,d^a=de,f^a=fh,c^b=cg,d^b=dh46 conjugacy classes — view the full character table.
| maximal subgroup of | #139515, #139516, #152756, … |
| maximal normal subgroup of | #139515, #139516, #152756, … |
| maximal quotient of | #297316, #297318, #297328, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 18723);