Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #18746 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#120 |
| commutator, Frattini, cosocle | C42 | C24 |
| prime radical | #3480 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
479 conjugacy classes of subgroups; only the 123 normal subgroups are shown.
This group has 479 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) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(2,7)(3,8)(4,5)(9,10,14,17)(11,18,20,23)(12,16,13,21)(15,22,24,19);(1,2)(3,4)(5,7)(6,8)(9,11)(10,15)(12,19)(13,22)(14,20)(16,23)(17,24)(18,21);(1,3,6,7)(2,4,8,5)(9,12,14,13)(10,16,17,21)(11,19,20,22)(15,23,24,18);(1,4,6,5)(2,3,8,7)Polycyclic presentation:
2^8|a^2=gh,c^2=g,d^2=h,e^2=g,f^2=h,b^a=be,c^a=cf,d^a=dh,e^a=eg,f^a=fh,e^b=eg,f^b=fh,e^c=eh,f^c=fh43 conjugacy classes — view the full character table.
| maximal subgroup of | #270629, #270630, #270653, … |
| maximal normal subgroup of | #270629, #270630, #270653, … |
| maximal quotient of | #166018, #166023, #166030, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 11791);