Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24468 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#120 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
795 conjugacy classes of subgroups; only the 143 normal subgroups are shown.
This group has 795 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) = 20 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 20 points:
(2,3)(5,6,9,12)(7,14,15,11)(8,10,13,19)(16,18,17,20);(1,2,4,3)(5,7,9,15)(6,11,12,14)(8,18,13,20)(10,16,19,17);(5,8,10,12,9,13,19,6)(7,11,16,18,15,14,17,20);(1,3,4,2)(6,12)(8,13)(11,14)(18,20)Polycyclic presentation:
2^8|a^2=h,b^2=e,c^2=f,d^2=eh,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=de,f^a=fh,c^b=cg,g^b=gh,d^c=dh,g^c=gh40 conjugacy classes — view the full character table.
| maximal subgroup of | #152734, #152735, #152738, … |
| maximal normal subgroup of | #152734, #152735, #152738, … |
| maximal quotient of | #297412, #297414, #297420, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 18740);