Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #61058 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ #532 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
1083 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2,4,6)(3,7,8,5)(9,10,13,18)(11,20,21,15)(12,17,23,24)(14,16,22,19);(1,3)(2,5)(4,8)(6,7)(9,11)(10,15)(12,14)(13,21)(16,24)(17,19)(18,20)(22,23);(10,16)(11,21)(14,22)(15,17)(18,19)(20,24);(3,8)(5,7)(9,12)(10,17)(11,22)(13,23)(14,21)(15,16)(18,24)(19,20);(10,18)(11,21)(12,23)(16,19)Polycyclic presentation:
2^8|a^2=f,g^2=h,b^a=bf,c^a=cg,e^a=eh,g^a=gh,c^b=ch,d^b=df,e^b=eh,g^c=gh,e^d=eh52 conjugacy classes — view the full character table.
| maximal subgroup of | #250531, #250535, #250563, … |
| maximal normal subgroup of | #250531, #250535, #250563, … |
| maximal quotient of | #6379100, #6379329, #6379723, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 54206);