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) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, cosocle | C23 | C25 |
| commutator | C22 | C24×C4 |
| prime radical | C24 | C24 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
1047 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2,4,3)(7,11)(9,12)(13,14,18,16)(15,21,23,19)(17,24,25,20)(22,26,28,27);(1,3,4,2)(5,6,8,10)(7,9,11,12)(14,16)(19,21)(20,24)(26,27);(5,7,8,11)(6,9,10,12)(13,15)(14,19)(16,21)(17,22)(18,23)(20,26)(24,27)(25,28);(13,16,18,14)(15,21,23,19)(17,24,25,20)(22,27,28,26);(13,17,18,25)(14,20,16,24)(15,22,23,28)(19,26,21,27)Polycyclic presentation:
2^8|a^2=h,b^2=gh,c^2=fg,d^2=f,e^2=f,b^a=bf,c^a=cg,e^a=ef,d^b=df88 conjugacy classes — view the full character table.
| maximal subgroup of | #105775, #105881, #106099, … |
| maximal normal subgroup of | #105775, #105881, #106099, … |
| maximal quotient of | #513553, #513571, #513597, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 27205);