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 | |
901 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2,5,4)(3,7,8,6)(11,18)(12,13)(15,23)(16,21)(19,24)(20,22);(1,3)(2,6)(4,7)(5,8)(9,10)(11,19)(12,21)(13,16)(14,17)(15,20)(18,24)(22,23);(1,4,5,2)(3,7,8,6)(9,11)(10,15)(12,18)(13,22)(14,20)(16,24)(17,19)(21,23);(3,8)(6,7)(9,10)(11,15)(12,21)(13,16)(14,17)(18,23)(19,20)(22,24);(9,12,14,13)(10,16,17,21)(11,18,20,22)(15,24,19,23)Polycyclic presentation:
2^8|a^2=f,c^2=f,e^2=h,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 | #6578034, #6578035, #6578036, … |
| maximal normal subgroup of | #6578034, #6578035, #6578036, … |
| maximal quotient of | #6379330, #6379331, #6379863, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 54282);