Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23848 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C2×#127 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×C4⋊C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
491 conjugacy classes of subgroups; only the 143 normal subgroups are shown.
This group has 491 conjugacy classes of subgroups — too many to draw. Only its 143 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:
(2,3)(5,6,7,8)(11,17)(14,23)(18,24)(19,22);(1,2,4,3)(9,10,12,15)(11,18,17,24)(13,21,20,16)(14,22,23,19);(6,8)(9,11,12,17)(10,14,15,23)(13,22,20,19)(16,24,21,18);(1,3,4,2)Polycyclic presentation:
2^8|a^2=fh,b^2=eh,c^2=h,d^2=e,g^2=h,b^a=be,c^a=cf,d^a=de,c^b=cg,g^b=gh,g^c=gh52 conjugacy classes — view the full character table.
| maximal subgroup of | #269733, #269750, #269763, … |
| maximal normal subgroup of | #269733, #269750, #269763, … |
| maximal quotient of | #276997, #276998, #276999, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 17721);