Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24412 |
| automorphism | Aut(G) order 32768 |
| 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 | #3496 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
657 conjugacy classes of subgroups; only the 129 normal subgroups are shown.
This group has 657 conjugacy classes of subgroups — too many to draw. Only its 129 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)(3,7)(4,6)(5,8)(9,10)(11,17)(12,20)(13,16)(14,19)(15,21)(18,23)(22,24);(1,3,5,4)(2,6,8,7)(11,18)(12,21)(15,20)(17,23);(1,3,5,4)(2,7,8,6)(9,11,12,19,13,18,21,24)(10,14,15,17,16,22,20,23);(1,4,5,3)(2,7,8,6)Polycyclic presentation:
2^8|b^2=e,c^2=ef,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 | #142653, #142655, #142749, … |
| maximal normal subgroup of | #142653, #142655, #142749, … |
| maximal quotient of | #303065, #303070, #303074, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 18857);