Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24412 |
| 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 | C22×C4 | C24 |
| prime radical | #532 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
479 conjugacy classes of subgroups; only the 125 normal subgroups are shown.
This group has 479 conjugacy classes of subgroups — too many to draw. Only its 125 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) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2,4,5)(3,7,8,6)(9,10,14,16)(11,15,19,24)(12,13,20,22)(17,21,23,18);(2,5)(6,7)(9,11,14,19)(10,15,16,24)(12,17,20,23)(13,21,22,18);(9,12,13,16,14,20,22,10)(11,17,18,15,19,23,21,24);(1,3,4,8)(2,6,5,7)Polycyclic presentation:
2^8|a^2=eh,b^2=h,c^2=f,d^2=e,f^2=h,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=gh,c^b=cg,f^b=fh,g^c=gh43 conjugacy classes — view the full character table.
| maximal subgroup of | #248278, #248283, #248338, … |
| maximal normal subgroup of | #248278, #248283, #248338, … |
| maximal quotient of | #295846, #295849, #295856, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 18684);