Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #45438 |
| automorphism | Aut(G) order 131072 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, prime radical, cosocle | C23 | C25 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
746 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 24 points:
(2,4)(6,7)(11,15)(14,16)(17,18,20,22)(19,23,24,21);(1,2,5,4)(3,7,8,6)(10,13)(11,15);(9,10)(11,14)(12,13)(15,16)(17,19,20,24)(18,21,22,23);(1,3,5,8)(2,6,4,7)(9,11,12,15)(10,14,13,16);(1,4,5,2)(3,7,8,6)(9,10)(11,14)(12,13)(15,16)Polycyclic presentation:
2^8|a^2=g,b^2=f,c^2=g,d^2=fh,e^2=f,b^a=bf,c^a=cg,d^a=dh,e^a=ef,c^b=ch,d^b=dfh,e^b=eh52 conjugacy classes — view the full character table.
| maximal subgroup of | #97197, #97336, #172365, … |
| maximal normal subgroup of | #97197, #97336, #172365, … |
| maximal quotient of | #557107, #557112, #557403, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 38994);