Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #7307 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C42 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #342 |
| commutator | C22×C4 | C42 |
| Frattini, cosocle | C23×Q8 | C22 |
| prime radical | C24 | C22⋊C4 |
| LCS | G ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C23×Q8 ◁ G | |
| LECS | G ▷ C23×Q8 ▷ C23 ▷ C2 ▷ C1 | |
259 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 2 generators on 24 points:
(1,2)(3,7)(4,6)(5,8)(9,10,12,16,13,17,21,24)(11,15,18,23,19,14,20,22);(1,3)(2,5,6,7)(4,8)(10,14,17,15)(16,23,24,22)Polycyclic presentation:
2^8|a^2=d,b^2=e,c^2=h,d^2=h,b^a=bc,c^a=cf,e^a=egh,f^a=fh,c^b=cg,d^b=dfh,d^c=dh34 conjugacy classes — view the full character table.
| maximal subgroup of | #93215, #93216, #93238, … |
| maximal normal subgroup of | #93215, #93216, #93238, … |
| maximal quotient of | #93110, #93114, #93117, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 296);