Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #18618 |
| automorphism | Aut(G) order 16384 |
| 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 | C2×#41 | D4 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
469 conjugacy classes of subgroups; only the 131 normal subgroups are shown.
This group has 469 conjugacy classes of subgroups — too many to draw. Only its 131 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:
(2,6)(3,4)(7,8)(9,10,13,16)(11,17,18,24)(12,20,21,15)(14,22,23,19);(1,2,5,8)(3,6,4,7)(12,21)(15,20)(17,24)(19,22);(9,11)(10,14)(12,22)(13,18)(15,24)(16,23)(17,20)(19,21);(1,3,5,4)(2,7,8,6)Polycyclic presentation:
2^8|a^2=h,b^2=g,d^2=g,e^2=g,f^2=h,b^a=be,c^a=cf,d^a=dg,e^a=eg,f^a=fh,d^b=dg,e^b=eg,f^b=fh,e^c=eh,f^c=fh46 conjugacy classes — view the full character table.
| maximal subgroup of | #245392, #245396, #245441, … |
| maximal normal subgroup of | #245392, #245396, #245441, … |
| maximal quotient of | #166514, #166521, #166530, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 11620);