Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30412 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | #2408 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×D4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
528 conjugacy classes of subgroups; only the 111 normal subgroups are shown.
This group has 528 conjugacy classes of subgroups — too many to draw. Only its 111 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)(11,13)(12,14)(15,16)(18,21)(22,23);(1,3)(2,5)(4,8)(6,7)(9,11,12,15)(10,13,14,16)(17,18)(19,21)(20,22)(23,24);(10,14)(13,16)(18,22)(20,24)(21,23);(2,6)(3,8)(10,14)(13,16)(18,21)(22,23)Polycyclic presentation:
2^8|b^2=f,g^2=h,b^a=be,c^a=cf,d^a=defh,c^b=cg,d^b=de,g^b=gh,g^c=gh40 conjugacy classes — view the full character table.
| maximal subgroup of | #142493, #142512, #143028, … |
| maximal normal subgroup of | #142493, #142512, #143028, … |
| maximal quotient of | #345888, #345889, #345900, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 24258);