Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #36702 |
| automorphism | Aut(G) = C215 (order 32768) · Inn(G) = C25 · Out(G) = C210 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C25 |
| prime radical | #2671 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
825 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2,4,5)(3,7,8,6)(12,16)(14,15)(19,23)(21,24);(2,5)(6,7)(10,11)(14,15)(17,18,20,22)(19,21,23,24);(9,10,13,11)(12,14,16,15)(17,19)(18,21)(20,23)(22,24);(1,2)(3,6)(4,5)(7,8)(9,11,13,10)(12,15,16,14);(1,3,4,8)(2,6,5,7)(9,12)(10,14)(11,15)(13,16)Polycyclic presentation:
2^8|a^2=f,b^2=g,c^2=h,d^2=h,e^2=f,b^a=bf,c^a=cg,d^a=df,e^a=efh,c^b=ch,d^b=dfh52 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#37238, #7634557, #7634565, … |
| maximal normal subgroup of | C2×#37238, #7634557, #7634565, … |
| maximal quotient of | #553740, #553749, #554031, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 30226);