Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #62253 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25 |
| prime radical | #3511 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C4×#41 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
877 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2)(3,10)(4,8)(5,14)(6,9)(7,12)(11,16)(13,15)(17,18,21,24)(19,25,26,32)(20,29,27,23)(22,30,31,28);(1,3)(2,7)(4,11)(5,13)(6,12)(8,15)(9,10)(14,16)(19,26)(22,31)(25,32)(28,30);(1,4,6,5)(2,8,9,14)(3,11,12,13)(7,15,10,16)(17,19,27,30,21,26,20,28)(18,22,29,32,24,31,23,25);(3,12)(7,10)(11,13)(15,16);(1,5,6,4)(2,8,9,14)(3,13,12,11)(7,15,10,16)Polycyclic presentation:
2^8|a^2=h,c^2=fgh,e^2=f,g^2=h,b^a=bf,c^a=cg,e^a=ef,g^a=gh,c^b=ch,d^b=df52 conjugacy classes — view the full character table.
| maximal subgroup of | #250255, #250257, #250259, … |
| maximal normal subgroup of | #250255, #250257, #250259, … |
| maximal quotient of | #6386944, #6389632, #6389639, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 55528);