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)(9,10,14,13,15,18,19,23)(11,16,12,17,20,22,21,24);(1,3)(2,5)(4,8)(6,7)(9,11)(10,16)(12,14)(13,17)(15,20)(18,22)(19,21)(23,24);(3,8)(5,7)(9,12)(10,17)(11,19)(13,22)(14,20)(15,21)(16,23)(18,24);(2,6)(5,7)(11,20)(12,21)(16,22)(17,24);(9,13,14,18,15,23,19,10)(11,17,12,22,20,24,21,16)Polycyclic presentation:
2^8|a^2=g,e^2=g,g^2=h,b^a=bf,d^a=df,e^a=eh,c^b=cfh,d^b=dh,d^c=dh76 conjugacy classes — view the full character table.
| maximal subgroup of | #6345417, #6345418, #6345427, … |
| maximal normal subgroup of | #6345417, #6345418, #6345427, … |
| maximal quotient of | #6370407, #6370412, #6370415, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 53307);