Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #47826 |
| automorphism | Aut(G) order 65536 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C25 |
| prime radical | #2631 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
827 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)(3,7)(4,5)(6,8)(9,10)(11,15)(12,14)(13,16)(17,18,22,23)(19,27,28,24)(20,25,26,21)(29,32,31,30);(2,5)(6,7)(9,10,13,16)(11,14,12,15)(18,23)(19,28)(21,25)(29,31);(9,11,13,12)(10,14,16,15)(17,19,22,28)(18,24,23,27)(20,31,26,29)(21,30,25,32);(1,2)(3,6)(4,5)(7,8)(9,10)(11,14)(12,15)(13,16)(17,20,22,26)(18,25,23,21)(19,29,28,31)(24,30,27,32);(1,3,4,8)(2,6,5,7)(9,12,13,11)(10,15,16,14)(17,21,22,25)(18,26,23,20)(19,30,28,32)(24,29,27,31)Polycyclic presentation:
2^8|a^2=h,b^2=gh,c^2=g,d^2=h,e^2=f,b^a=bf,c^a=cg,e^a=ef,c^b=ch,d^b=dfh,e^b=eh,d^c=dh52 conjugacy classes — view the full character table.
| maximal subgroup of | #132486, #253530, #253561, … |
| maximal normal subgroup of | #132486, #253530, #253561, … |
| maximal quotient of | #560387, #560526, #561571, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 44212);