Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3532 |
| automorphism | Aut(G) = C23×#1202 (order 8192) · Inn(G) = C25 · Out(G) = C23×H(5,𝔽2) |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C2×C4 | C25 |
| commutator, socle | C22 | C24×C4 |
| prime radical | #3475 | C2 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
913 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 24 points:
(2,5)(3,7)(9,10,13,16,12,14,20,21)(11,17,19,24,18,15,22,23);(1,2)(3,6)(4,5)(7,8)(10,14)(15,17)(16,21)(23,24);(2,5)(6,8)(10,14)(15,17)(16,21)(23,24);(1,3,4,7)(2,6,5,8)(9,11,12,18)(10,15,14,17)(13,19,20,22)(16,23,21,24);(11,18)(15,17)(19,22)(23,24)Polycyclic presentation:
2^8|a^2=g,d^2=f,g^2=h,b^a=bf,c^a=ch,d^a=df,c^b=cfh,e^d=eh76 conjugacy classes — view the full character table.
| maximal subgroup of | #129708, #210274, #210275, … |
| maximal normal subgroup of | #129708, #210274, #210275, … |
| maximal quotient of | #6370218, #6370230, #6370299, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 53261);