Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24412 |
| 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 | C23 | C22×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
467 conjugacy classes of subgroups; only the 145 normal subgroups are shown.
This group has 467 conjugacy classes of subgroups — too many to draw. Only its 145 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:
(1,2,5,8)(3,7,4,6)(9,10,14,18)(11,19,22,16)(12,15,17,13)(20,24,23,21);(1,3,5,4)(2,6,8,7)(9,11)(10,16)(12,23)(13,21)(14,22)(15,24)(17,20)(18,19);(9,12,15,18,14,17,13,10)(11,20,21,19,22,23,24,16);(1,4,5,3)(2,7,8,6)(9,13,14,15)(10,17,18,12)(11,21,22,24)(16,20,19,23)Polycyclic presentation:
2^8|a^2=e,b^2=eh,c^2=f,d^2=e,f^2=h,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=gh,c^b=cg,f^b=fh,g^c=gh43 conjugacy classes — view the full character table.
| maximal subgroup of | #142692, #142694, #142788, … |
| maximal normal subgroup of | #142692, #142694, #142788, … |
| maximal quotient of | #302027, #302036, #302041, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 18813);