Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2403 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C22×C4 | C24 |
| commutator, socle, prime radical | C23 | C23×C4 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
353 conjugacy classes of subgroups; only the 175 normal subgroups are shown.
This group has 353 conjugacy classes of subgroups — too many to draw. Only its 175 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) = 256 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2)(3,6)(4,5)(7,8)(10,14)(11,12)(17,18,21,22,20,24,19,23);(9,10,13,14)(11,16,12,15)(17,19,20,21)(18,22,24,23);(2,5)(6,8)(9,11,13,12)(10,15,14,16)(17,19,20,21)(18,23,24,22);(1,3,4,7)(2,6,5,8)(9,12,13,11)(10,15,14,16)(18,24)(22,23)Polycyclic presentation:
2^8|a^2=g,b^2=e,c^2=e,d^2=f,g^2=h,b^a=be,c^a=cf,d^a=de,c^b=ceh76 conjugacy classes — view the full character table.
| maximal subgroup of | #164371, #164375, #164393, … |
| maximal normal subgroup of | #164371, #164375, #164393, … |
| maximal quotient of | #355321, #355375, #355563, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 15008);