Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #61071 |
| 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 |
| prime radical | #3511 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ #532 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
898 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)(10,15)(11,18)(14,17)(16,19)(20,22)(23,24);(1,3)(2,6)(4,7)(5,8)(9,10)(11,16)(12,14)(13,15)(17,21)(18,24)(19,22)(20,23);(1,4,5,2)(3,7,8,6)(9,11)(10,16)(12,19)(13,20)(14,22)(15,23)(17,18)(21,24);(3,8)(6,7)(9,12)(10,17)(11,19)(13,21)(14,15)(16,18)(20,24)(22,23);(10,15)(12,21)(16,23)(19,24)Polycyclic presentation:
2^8|a^2=fh,c^2=fh,g^2=h,b^a=bf,c^a=cg,g^a=gh,d^b=df,e^b=eh,g^c=gh,e^d=eh55 conjugacy classes — view the full character table.
| maximal subgroup of | #250081, #250113, #250145, … |
| maximal normal subgroup of | #250081, #250113, #250145, … |
| maximal quotient of | #6379319, #6379323, #6379876, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 54270);