Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3532 |
| automorphism | Aut(G) = C27×#120 (order 4096) · Inn(G) = C25 · Out(G) = C24×D4 |
| 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 | #3460 | C2 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
924 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2)(3,7)(4,6)(5,8)(9,10,14,18,13,16,21,24)(11,12,17,20,19,15,22,23);(1,3)(2,5)(4,8)(6,7)(9,11)(10,15)(12,16)(13,19)(14,17)(18,23)(20,24)(21,22);(3,8)(5,7)(11,19)(12,15)(17,22)(20,23);(2,6)(5,7)(10,16)(11,19)(17,22)(18,24);(9,12,14,20,13,15,21,23)(10,17,18,19,16,22,24,11)Polycyclic presentation:
2^8|a^2=g,e^2=g,g^2=h,b^a=bf,d^a=df,c^b=cf,d^b=dh,e^b=eh,e^c=eh76 conjugacy classes — view the full character table.
| maximal subgroup of | #6345421, #6345422, #6345431, … |
| maximal normal subgroup of | #6345421, #6345422, #6345431, … |
| maximal quotient of | #6370412, #6370415, #6370424, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 53314);