Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4×#120 |
| automorphism | Aut(G) order 65536 |
| 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 | C22×D4 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
866 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2,5,7)(3,4,6,8)(10,15)(11,16)(18,22)(23,24);(1,3)(2,4)(5,6)(7,8)(9,10)(11,12)(13,16)(14,15)(17,18,21,22)(19,23,20,24);(1,4)(2,6)(3,7)(5,8)(9,11,14,16)(10,12,15,13)(17,19,21,20)(18,23,22,24);(1,3)(2,4)(5,6)(7,8)(9,12,14,13)(10,11,15,16)(17,19,21,20)(18,23,22,24);(1,4)(2,6)(3,7)(5,8)(9,13,14,12)(10,16,15,11)(17,20,21,19)(18,24,22,23)Polycyclic presentation:
2^8|a^2=h,b^2=fg,c^2=f,d^2=f,e^2=f,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 | #6580288, #6580293, #6580353, … |
| maximal normal subgroup of | #6580288, #6580293, #6580353, … |
| maximal quotient of | #519008, #519032, #519077, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 29166);