Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4×#120 |
| 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, cosocle | C23 | C25 |
| prime radical | C24 | C24 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
932 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2,5,3)(4,6,8,7)(9,10)(11,12)(13,16)(14,15);(1,2,5,3)(4,7,8,6)(10,15)(11,13)(17,18,22,20)(19,24,21,23);(2,3)(6,7)(9,11,14,13)(10,12,15,16)(17,19,22,21)(18,23,20,24);(1,3,5,2)(4,6,8,7)(9,12)(10,11)(13,15)(14,16)(17,20,22,18)(19,23,21,24);(1,4,5,8)(2,6,3,7)(9,13,14,11)(10,16,15,12)(17,21,22,19)(18,24,20,23)Polycyclic presentation:
2^8|a^2=g,b^2=fh,c^2=gh,d^2=fh,e^2=h,b^a=bf,c^a=cg,c^b=ch,e^b=eh,d^c=dh,e^d=eh70 conjugacy classes — view the full character table.
| maximal subgroup of | #98199, #224131, #224137, … |
| maximal normal subgroup of | #98199, #224131, #224137, … |
| maximal quotient of | #519104, #519115, #519125, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 29552);