Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30484 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | D42 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C23 | H(5,𝔽2) |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
327 conjugacy classes of subgroups; only the 115 normal subgroups are shown.
This group has 327 conjugacy classes of subgroups — too many to draw. Only its 115 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:
(1,2,5,7)(3,4,8,6)(9,10,13,18,14,19,12,17)(11,15,22,16,21,24,20,23);(1,3,5,8)(2,6,7,4)(9,11)(10,15)(12,20)(13,22)(14,21)(16,18)(17,23)(19,24);(9,10,14,19)(11,16,21,23)(12,18,13,17)(15,20,24,22);(1,4,5,6)(2,3,7,8)(9,11)(10,16)(12,22)(13,20)(14,21)(15,18)(17,24)(19,23)Polycyclic presentation:
2^8|a^2=efh,b^2=e,c^2=h,d^2=e,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=def,c^b=cg,d^b=de,g^b=gh,f^c=fh,g^c=gh,f^d=fh40 conjugacy classes — view the full character table.
| maximal subgroup of | #125734, #125736, #125948, … |
| maximal normal subgroup of | #125734, #125736, #125948, … |
| maximal quotient of | #345524, #345525, #345529, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 24244);