Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23860 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C2×#547 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×D4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
877 conjugacy classes of subgroups; only the 153 normal subgroups are shown.
This group has 877 conjugacy classes of subgroups — too many to draw. Only its 153 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2)(3,4)(9,10,15,19)(11,20,12,17)(13,18,14,23)(16,21,24,22);(5,6)(7,8)(9,11,15,12)(10,16,19,24)(13,22,14,21)(17,23,20,18);(2,4)(6,8)(9,12,15,11)(10,17,19,20)(13,22,14,21)(16,23,24,18);(9,13)(11,21)(12,22)(14,15)(16,24)(18,23)Polycyclic presentation:
2^8|a^2=h,b^2=h,c^2=h,e^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg,d^b=dh,e^b=eh,e^d=eh46 conjugacy classes — view the full character table.
| maximal subgroup of | #139226, #139286, #267961, … |
| maximal normal subgroup of | #139226, #139286, #267961, … |
| maximal quotient of | #244934, #244950, #244965, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 16853);