Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24464 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#120 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | #3038 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
569 conjugacy classes of subgroups; only the 131 normal subgroups are shown.
This group has 569 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) = 16 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 16 points:
(1,2,5,8)(3,7,4,6)(9,10)(11,14)(12,15)(13,16);(1,3,5,4)(2,6,8,7)(9,11,14,10)(12,13,16,15);(1,3,5,4)(2,7,8,6)(9,12,13,11,14,16,15,10);(1,4,5,3)(2,7,8,6)(9,13,14,15)(10,12,11,16)Polycyclic presentation:
2^8|a^2=eh,b^2=e,c^2=ef,d^2=e,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=de,f^a=fh,c^b=cg,g^b=gh,g^c=gh46 conjugacy classes — view the full character table.
| maximal subgroup of | #244656, #244666, #244671, … |
| maximal normal subgroup of | #244656, #244666, #244671, … |
| maximal quotient of | #304310, #304312, #304313, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 18912);