Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9986 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #392 |
| commutator, Frattini, cosocle | C23×C4 | C23 |
| prime radical | C2×#121 | C22 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
362 conjugacy classes of subgroups; only the 53 normal subgroups are shown.
This group has 362 conjugacy classes of subgroups — too many to draw. Only its 53 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 — 3 generators on 24 points:
(1,2)(3,5)(4,6)(7,8)(9,10,14,18)(11,19,22,24)(12,16,13,17)(15,21,23,20);(5,8)(9,11,14,22)(10,15,18,23)(12,21,13,20)(16,19,17,24);(1,3)(2,5)(4,7)(6,8)(10,16)(11,20)(12,13)(17,18)(19,24)(21,22)Polycyclic presentation:
2^8|a^2=g,b^2=g,d^2=g,e^2=g,f^2=g,b^a=bd,c^a=ce,d^a=dg,e^a=eg,f^a=fgh,c^b=cf,d^b=dg,e^b=eg,f^b=fg,d^c=dh,e^c=eg,f^c=fg34 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#9989, #232574, #232582, … |
| maximal normal subgroup of | C2×#9989, #232574, #232582, … |
| maximal quotient of | #102472, #102481, #102486, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 2977);