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 | #3480 | C2 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
909 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(2,5)(3,7)(9,10,12,16,13,17,20,24)(11,14,19,21,18,15,22,23);(1,2)(3,6)(4,5)(7,8)(9,11)(10,14)(12,19)(13,18)(15,17)(16,21)(20,22)(23,24);(2,5)(6,8)(9,11)(10,15)(12,19)(13,18)(14,17)(16,23)(20,22)(21,24);(1,3,4,7)(2,6,5,8)(9,11)(10,14)(12,19)(13,18)(15,17)(16,21)(20,22)(23,24);(11,18)(14,15)(19,22)(21,23)Polycyclic presentation:
2^8|a^2=g,d^2=f,g^2=h,b^a=bf,c^a=ch,d^a=df,c^b=cf,e^b=eh,e^c=eh,e^d=eh76 conjugacy classes — view the full character table.
| maximal subgroup of | #129712, #210270, #210271, … |
| maximal normal subgroup of | #129712, #210270, #210271, … |
| maximal quotient of | #6370229, #6370230, #6370309, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 53263);