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 | #2703 | 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:
(4,8)(6,7)(10,14)(11,12)(19,23)(21,24);(2,3)(6,7)(9,10,13,14)(11,15,12,16)(17,18,20,22)(19,21,23,24);(1,2,5,3)(4,6,8,7)(9,11,13,12)(10,15,14,16)(17,19)(18,21)(20,23)(22,24);(1,3,5,2)(4,7,8,6)(9,10)(11,15)(12,16)(13,14);(1,4)(2,6)(3,7)(5,8)(9,12,13,11)(10,16,14,15)Polycyclic presentation:
2^8|b^2=g,c^2=fh,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×#37239, #7634562, #7634563, … |
| maximal normal subgroup of | C2×#37239, #7634562, #7634563, … |
| maximal quotient of | #553740, #553749, #554031, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 30227);