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 | C25 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
996 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2,5,3)(4,6,8,7)(9,10,13,15)(11,12,14,16)(18,23)(19,24);(1,2,5,3)(4,7,8,6)(9,11)(10,12)(13,14)(15,16)(17,18)(19,20)(21,24)(22,23);(2,3)(6,7)(9,12)(10,14)(11,15)(13,16)(17,19,22,24)(18,20,23,21);(1,3,5,2)(4,6,8,7)(9,11)(10,12)(13,14)(15,16)(17,20,22,21)(18,19,23,24);(1,4)(2,6)(3,7)(5,8)(9,12)(10,14)(11,15)(13,16)(17,21,22,20)(18,24,23,19)Polycyclic presentation:
2^8|a^2=h,b^2=fg,c^2=f,d^2=g,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 | #223023, #223043, #223065, … |
| maximal normal subgroup of | #223023, #223043, #223065, … |
| maximal quotient of | #519178, #519190, #519246, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 29195);