Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23391 |
| automorphism | Aut(G) order 262144 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C22×#41 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1009 conjugacy classes of subgroups; only the 199 normal subgroups are shown.
This group has 1009 conjugacy classes of subgroups — too many to draw. Only its 199 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:
(1,2)(3,4)(9,10)(11,20)(12,23)(13,17)(14,18)(15,19)(16,21)(22,24);(2,4)(5,6)(7,8)(9,11,15,12)(10,16,19,24)(13,22,14,21)(17,20,18,23);(6,8)(9,12,15,11)(10,16,19,24)(13,21,14,22)(17,20,18,23);(9,13,15,14)(10,17,19,18)(11,21,12,22)(16,23,24,20)Polycyclic presentation:
2^8|b^2=h,c^2=h,d^2=h,e^2=h,f^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fh,c^b=cg,d^b=dh,e^b=eh,f^b=fh,d^c=dh,e^c=eh,f^c=fh52 conjugacy classes — view the full character table.
| maximal subgroup of | #101263, #101266, #150185, … |
| maximal normal subgroup of | #101263, #101266, #150185, … |
| maximal quotient of | #232237, #232242, #232251, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 16420);