Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #18376 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#120 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C22×D4 | D4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
505 conjugacy classes of subgroups; only the 129 normal subgroups are shown.
This group has 505 conjugacy classes of subgroups — too many to draw. Only its 129 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) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(2,5)(3,7)(6,8)(9,10,14,18)(11,19,22,23)(12,17,13,16)(15,20,24,21);(1,2,4,6)(3,8,7,5)(9,11,14,22)(10,15,18,24)(12,21,13,20)(16,19,17,23);(3,7)(5,8);(2,6)(5,8)(9,12,14,13)(10,16,18,17)(11,20,22,21)(15,23,24,19)Polycyclic presentation:
2^8|a^2=gh,b^2=g,d^2=gh,e^2=g,b^a=be,c^a=cf,d^a=dgh,e^a=eg,d^b=dg,e^b=eg,f^b=fh,e^c=eh43 conjugacy classes — view the full character table.
| maximal subgroup of | #142770, #142771, #142786, … |
| maximal normal subgroup of | #142770, #142771, #142786, … |
| maximal quotient of | #169652, #169716, #169780, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 11385);