Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3484 |
| automorphism | Aut(G) order 524288 |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C210 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, cosocle | C23 | C25 |
| commutator | C22 | C24×C4 |
| prime radical | C22×H(5,𝔽2) | C2 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
2949 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 16 points:
(2,4)(5,6,8,7)(9,10)(11,15)(12,13)(14,16);(10,13)(14,15);(1,2,3,4)(5,7,8,6);(9,10)(11,14)(12,13)(15,16);(9,11)(10,14)(12,16)(13,15)Polycyclic presentation:
2^8|a^2=h,c^2=gh,b^a=bf,c^a=cg,e^a=ef,d^b=df88 conjugacy classes — view the full character table.
| maximal subgroup of | #96962, #97052, #129221, … |
| maximal normal subgroup of | #96962, #97052, #129221, … |
| maximal quotient of | #513573, #513574, #513736, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 27244);